Integral Calculus is one of the most important and high-scoring units in NIOS Class 12 Senior Secondary Mathematics Course 315. This chapter helps students understand integration techniques, standard formulas, definite integrals, and applications of integration.
This blog provides step-by-step solutions, shortcuts, important formulas, and exam-oriented guidance for solving Integral Calculus Terminal Exercise problems easily.
Whether you are preparing for:
- NIOS board examinations
- Assignment submission
- Quick revision
- Competitive exam basics
- Self-study learning
these solutions will help improve both speed and conceptual understanding.
Why Integral Calculus Is Important
Integral Calculus is widely used in:
- Physics
- Engineering
- Statistics
- Area calculations
- Motion problems
- Differential equations
Students can score high marks in exams by mastering standard integration methods and formulas.
Terminal exercise
Question no: 1
\begin{array}{l} {\int{\sqrt{{1}\mathrm{{+}}\sin{2}{x}}}\hspace{0.33em}{dx}}\\ {{1}\mathrm{{+}}\sin{2}{x}\mathrm{{=}}{\sin}^{2}{x}\mathrm{{+}}{\cos}^{2}{x}\mathrm{{+}}{\sin}^{2}{x}}\\ {\mathrm{{=}}{\mathrm{(}}\sin{x}\mathrm{{+}}\cos{x}{\mathrm{)}}^{2}}\\ {\int{\sqrt{{1}\mathrm{{+}}\sin{2}{x}}}\hspace{0.33em}{dx}}\\ {\mathrm{{=}}\int{\sqrt{{\left({\sin{x}\hspace{0.33em}\mathrm{{+}}\hspace{0.33em}\cos{x}}\right)}^{2}}}\hspace{0.33em}{dx}}\\ {\mathrm{{=}}\int{\sin{x}\hspace{0.33em}{dx}\mathrm{{+}}\int{\cos\hspace{0.33em}{x}}}\hspace{0.33em}{dx}}\\ {\mathrm{{=}}\mathrm{{-}}\cos{x}\mathrm{{+}}\sin{x}\mathrm{{+}}{C}}\\ {\mathrm{{=}}\sin{x}\mathrm{{-}}\cos{x}\hspace{0.33em}\mathrm{{+}}\hspace{0.33em}{C}\hspace{0.33em}}\\ {{where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}Question no: 3
\begin{array}{l} {\int{\frac{\cos2x}{{\cos}^{2}{x}\hspace{0.33em}{\sin}^{2}{x}}}dx}\\ {\mathrm{{=}}\int{\frac{{\cos}^{2}{x}\mathrm{{-}}{\sin}^{2}{x}}{{\cos}^{2}{x}\hspace{0.33em}{\sin}^{2}{x}}}{dx}}\\ {\mathrm{{=}}\int{\frac{1}{{\sin}^{2}x}}{dx}\mathrm{{-}}\int{\frac{1}{{\cos}^{2}x}dx}}\\ {\mathrm{{=}}\int{{Co}{\sec}^{2}{x}\hspace{0.33em}{dx}}\mathrm{{-}}\int{{\sec}^{2}{x}\hspace{0.33em}{dx}}}\\ {{Ans}\mathrm{{=}}\mathrm{{-}}\cot{x}\mathrm{{-}}\tan{x}\mathrm{{+}}{C}}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}
Question no: 4
\begin{array}{l} {{\int{\left({\tan{x}\mathrm{{-}}\cot{x}}\right)}}^{2}dx}\\ {\mathrm{{=}}\int{{\mathrm{(}}{\tan}^{2}{x}\mathrm{{+}}{\cot}^{2}{x}}\mathrm{{-}}{2}\tan{x}\hspace{0.33em}\cot{x}{\mathrm{)}}{dx}}\\ {\mathrm{{=}}\int{\left({{\sec}^{2}{x}\mathrm{{-}}{1}\mathrm{{+}}{Co}{\sec}^{2}{x}\mathrm{{-}}{1}\mathrm{{-}}{2}\tan{x}\cot{x}}\right)}{dx}}\\ {\mathrm{{=}}\int{{\sec}^{2}{x}\hspace{0.33em}{dx}\mathrm{{+}}\int{{Co}{\sec}^{2}{x}\hspace{0.33em}{dx}}}\mathrm{{-}}{2}\int{dx}\mathrm{{-}}{2}\int{\tan{x}\hspace{0.33em}\frac{1}{\tan{x}}\hspace{0.33em}{dx}}}\\ {{Ans}\mathrm{{=}}\tan{x}\mathrm{{-}}\cot{x}\mathrm{{-}}{4}{x}\mathrm{{+}}{C}}\\ {{Where}{\mathrm{,}}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}Question no: 5
\begin{array}{l} {\int{\left({\frac{4}{{1}\mathrm{{+}}{x}^{2}}\mathrm{{-}}\frac{1} {\sqrt{{1}\mathrm{{-}}{x}^{2}}}}\right)}dx}\\ {}\\ {\mathrm{{=}}\int{\frac{4}{{1}\mathrm{{+}}{x}^{2}}{dx}\mathrm{{-}}\int{\frac{1}{\sqrt{{1}\mathrm{{-}}{x}^{2}}}dx}}}\\ {}\\ {\mathrm{{=}}{4}\int{\frac{1}{{1}\mathrm{{+}}{x}^{2}}{dx}\mathrm{{-}}\int{\frac{1}{\sqrt{{1}\mathrm{{-}}{x}^{2}}}}{dx}}}\\ {}\\ {{Ans}\mathrm{{=}}{4}{\tan}^{\mathrm{{-}}{1}}{x}\mathrm{{-}}{\sin}^{\mathrm{{-}}{1}}{x}\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}
Question no: 6
\begin{array}{l} {\int{\frac{2{\sin}^{2}x}{{1}\mathrm{{+}}\cos{2}{x}}dx}}\\ {}\\ {\mathrm{{=}}\int{\frac{2{\sin}^{2}x}{{1}\mathrm{{+}}{1}\mathrm{{-}}{2}{\sin}^{2}{x}}dx}}\\ {}\\ {\mathrm{{=}}\int{\frac{2{\sin}^{2}x}{{2}\mathrm{{-}}{2}{\sin}^{2}{x}}}{dx}}\\ {}\\ {\mathrm{{=}}\int{\frac{{\sin}^{2}x}{{1}\mathrm{{-}}{\sin}^{2}{x}}dx}}\\ {}\\ {\mathrm{{=}}\int{\frac{{\sin}^{2}x}{{\cos}^{2}x}dx}}\\ {}\\ {\mathrm{{=}}\int{{\tan}^{2}{x}\hspace{0.33em}{dx}}}\\ {}\\ {\mathrm{{=}}\int{{\mathrm{(}}{\sec}^{2}{x}\mathrm{{-}}{1}{\mathrm{)}}}{dx}}\\ {}\\ {\mathrm{{=}}\tan{x}\mathrm{{-}}{x}\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}Question no: 7
\begin{array}{l} {\int{\frac{2{\cos}^{2}x}{{1}\mathrm{{-}}\cos{2}{x}}dx}}\\ {}\\ {\mathrm{{=}}\int{\frac{2{\cos}^{2}x}{{1}\mathrm{{-}}{\mathrm{(}}{1}\mathrm{{-}}{2}{\sin}^{2}{x}{\mathrm{)}}}dx}}\\ {}\\ {\mathrm{{=}}\int{\frac{2{\cos}^{2}x}{\mathrm{{-}}{2}{\sin}^{2}{x}}dx}}\\ {}\\ {\mathrm{{=}}\mathrm{{-}}\int{\frac{{\cos}^{2}x}{{\sin}^{2}x}dx}}\\ {}\\ {\mathrm{{=}}\mathrm{{-}}\int{{\cot}^{2}{x}\hspace{0.33em}{dx}}}\\ {}\\ {\mathrm{{=}}\mathrm{{-}}\int{{\mathrm{(}}{Co}{\sec}^{2}{x}\mathrm{{-}}{1}{\mathrm{)}}}{dx}}\\ {}\\ {\mathrm{{=}}\mathrm{{-}}\int{{Co}{\sec}^{2}{x}\hspace{0.33em}{dx}\hspace{0.33em}\mathrm{{-}}\int{dx}}}\\ {}\\ {{Ans}\mathrm{{=}}\mathrm{{-}}\cot{x}\mathrm{{-}}{x}\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}Question no: 8
\begin{array}{l} {\int{{\left({\sin\frac{x}{2}\mathrm{{+}}\cos\frac{x}{2}}\right)}^{2}dx}}\\ {}\\ {{I}\mathrm{{=}}\int{\left({{\sin}^{2}\left({\frac{x}{2}}\right)\mathrm{{+}}{\cos}^{2}\left({\frac{x}{2}}\right)\mathrm{{+}}{2}\sin\frac{x}{2}\cos\frac{x}{2}}\right){dx}}}\\ {}\\ {{By}\hspace{0.33em}{Pythogorean}\hspace{0.33em}{identity}{\mathrm{,}}}\\ {}\\ {{\sin}^{2}{x}\mathrm{{+}}{\cos}^{2}{x}\mathrm{{=}}{1}}\\ {}\\ {{I}\mathrm{{=}}\int{\left({{1}\mathrm{{+}}{2}\sin\frac{x}{2}\cos\frac{x}{2}}\right){dx}}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\int{{\mathrm{(}}{1}\mathrm{{+}}\sin{x}{\mathrm{)}}}\hspace{0.33em}{dx}}\\ {}\\ {{2}\sin{x}\hspace{0.33em}\cos{x}\mathrm{{=}}\sin{2}{x}}\\ {}\\ {{2}\sin\frac{x}{2}\cos\frac{x}{2}\mathrm{{=}}\sin{x}}\\ {}\\ {{I}\mathrm{{=}}\int{\left({{1}\mathrm{{+}}\sin{x}}\right)}{dx}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\int{{dx}\mathrm{{+}}\int{\sin{x}\hspace{0.33em}{dx}}}}\\ {}\\ {{Ans}\mathrm{{=}}{x}\mathrm{{-}}\cos{x}\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}
Question no: 9
\begin{array}{l} {\int{{\left({\sin\frac{x}{2}\mathrm{{-}}\cos\frac{x}{2}}\right)}^{2}dx}}\\ {}\\ {{I}\mathrm{{=}}\int{\left({{\sin}^{2}\left({\frac{x}{2}}\right)\mathrm{{+}}{\cos}^{2}\left({\frac{x}{2}}\right)\mathrm{{-}}{2}\sin\frac{x}{2}\cos\frac{x}{2}}\right){dx}}}\\ {}\\ {{By}\hspace{0.33em}{Pythogorean}\hspace{0.33em}{identity}{\mathrm{,}}}\\ {}\\ {{\sin}^{2}{x}\mathrm{{+}}{\cos}^{2}{x}\mathrm{{=}}{1}}\\ {}\\ {{I}\mathrm{{=}}\int{\left({{1}\mathrm{{-}}{2}\sin\frac{x}{2}\cos\frac{x}{2}}\right){dx}}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\int{{\mathrm{(}}{1}\mathrm{{-}}\sin{x}{\mathrm{)}}}\hspace{0.33em}{dx}}\\ {}\\ {{2}\sin{x}\hspace{0.33em}\cos{x}\mathrm{{=}}\sin{2}{x}}\\ {}\\ {{2}\sin\frac{x}{2}\cos\frac{x}{2}\mathrm{{=}}\sin{x}}\\ {}\\ {{I}\mathrm{{=}}\int{\left({{1}\mathrm{{-}}\sin{x}}\right)}{dx}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\int{{dx}\mathrm{{-}}\int{\sin{x}\hspace{0.33em}{dx}}}}\\ {}\\ {{Ans}\mathrm{{=}}{x}\mathrm{{+}}\cos{x}\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}
Question no: 10
\begin{array}{l} {\int{\cos\left({{7}{x}\mathrm{{-}}\mathit{\pi}}\right)}dx}\\ {}\\ {{Let}\hspace{0.33em}{t}\mathrm{{=}}{7}{x}\mathrm{{-}}\mathit{\pi}}\\ {}\\ {{dt}\mathrm{{=}}{7}{dx}\hspace{0.33em}\mathrm{\Rightarrow}\hspace{0.33em}{dx}\mathrm{{=}}\frac{dt}{7}}\\ {}\\ {\int{\cos\left({{7}{x}\mathrm{{-}}\mathit{\pi}}\right)}\hspace{0.33em}{dx}\mathrm{{=}}\int{\frac{1}{7}\cos{t}\hspace{0.33em}{dt}}}\\ {}\\ {\mathrm{{=}}\frac{1}{7}\sin\hspace{0.33em}{t}}\\ {}\\ {{Ans}\mathrm{{=}}\frac{1}{7}\sin\left({{7}{x}\mathrm{{-}}\mathit{\pi}}\right)\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}Question no: 11
\begin{array}{l} {\int{\sin\left({{3}{x}\mathrm{{+}}{4}}\right)}dx}\\ {}\\ {{Let}\hspace{0.33em}{t}\mathrm{{=}}{3}{x}\mathrm{{+}}{4}}\\ {}\\ {{dt}\mathrm{{=}}{3}{dx}\hspace{0.33em}\mathrm{\Rightarrow}\hspace{0.33em}{dx}\mathrm{{=}}\frac{dt}{3}}\\ {}\\ {\int{\sin\left({{3}{x}\mathrm{{+}}{4}}\right)}\hspace{0.33em}{dx}\mathrm{{=}}\int{\frac{1}{3}\sin{t}\hspace{0.33em}{dt}}}\\ {}\\ {\mathrm{{=}}\mathrm{{-}}\frac{1}{3}\cos\hspace{0.33em}{t}}\\ {}\\ {{Ans}\mathrm{{=}}\mathrm{{-}}\frac{1}{3}\cos\left({{3}{x}\mathrm{{+}}{4}}\right)\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}
Question no: 12
\begin{array}{l} {\int{{\cos}^{2}\left({{2}{x}\mathrm{{+}}{b}}\right)}dx}\\ {}\\ {{Let}\hspace{0.33em}{t}\mathrm{{=}}{2}{x}\mathrm{{+}}{b}}\\ {}\\ {{dt}\mathrm{{=}}{2}{dx}\mathrm{\Rightarrow}{dx}\mathrm{{=}}\frac{dt}{2}}\\ {}\\ {\int{{\cos}^{2}\left({{2}{x}\mathrm{{+}}{b}}\right)}{dx}\mathrm{{=}}\frac{1}{2}\int{{\cos}^{2}{t}\hspace{0.33em}{dt}}}\\ {}\\ {\mathrm{{=}}\frac{1}{2}\int{\frac{{1}\mathrm{{+}}\cos{2}{t}}{2}}\hspace{0.33em}{dt}}\\ {}\\ {\mathrm{{=}}\frac{1}{4}\int{\left({{1}\mathrm{{+}}\cos{2}{t}}\right)}{dt}}\\ {}\\ {\mathrm{{=}}\frac{t}{4}\mathrm{{+}}\frac{\sin2t}{{2}\left({4}\right)}}\\ {}\\ {{Ans}\mathrm{{=}}\frac{{2}{x}\mathrm{{+}}{b}}{4}\mathrm{{+}}\frac{\sin\left({{4}{x}\mathrm{{+}}{2}{b}}\right)}{8}\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}
Question no: 14
\begin{array}{l} {\int{\frac{dx}{\left({{1}\mathrm{{+}}{x}^{2}}\right){\tan}^{\mathrm{{-}}{1}}{x}}}}\\ {}\\ {{I}\mathrm{{=}}\int{\frac{dx}{\left({{1}\mathrm{{+}}{x}^{2}}\right){\tan}^{\mathrm{{-}}{1}}}}}\\ {}\\ {{Let}\hspace{0.33em}{u}\mathrm{{=}}{\tan}^{\mathrm{{-}}{1}}{x}}\\ {}\\ {\mathrm{\Rightarrow}{du}\mathrm{{=}}\frac{dx}{{1}\mathrm{{+}}{x}^{2}}}\\ {}\\ {{I}\mathrm{{=}}\int{\frac{du}{u}}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\log\left|{u}\right|}\\ {}\\ {{Ans}\mathrm{{=}}\log\left|{{\tan}^{\mathrm{{-}}{1}}x}\right|\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}} \end{array}Question no: 15
\begin{array}{l} {\int{\frac{{Co}\sec\hspace{0.33em}{x}}{\log\left({\tan\frac{x}{2}}\right)}}dx}\\ {}\\ {Solution}\\ {}\\ {{I}\mathrm{{=}}\int{\frac{{Co}\sec\hspace{0.33em}{x}}{\log\left({\tan\frac{x}{2}}\right)}}{dx}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{\longrightarrow}\left({1}\right)}\\ {}\\ {{Let}\hspace{0.33em}{u}\mathrm{{=}}\frac{x}{2}}\\ {}\\ {\mathrm{\Rightarrow}\frac{du}{dx}\mathrm{{=}}\frac{1}{2}}\\ {}\\ {\mathrm{\Rightarrow}{dx}\mathrm{{=}}{2}{du}}\\ {}\\ {\left({1}\right)\hspace{0.33em}{becomes}{\mathrm{,}}}\\ {}\\ {{I}\mathrm{{=}}{2}\int{\frac{{co}\sec2u}{\log{\mathrm{(}}\tan\hspace{0.33em}{u}{\mathrm{)}}}}{du}\hspace{0.33em}\hspace{0.33em}\mathrm{\longrightarrow}\left({2}\right)}\\ {}\\ {{By}\hspace{0.33em}{the}\hspace{0.33em}{method}\hspace{0.33em}{of}\hspace{0.33em}{substitution}}\\ {}\\ {{Co}\sec{2}{u}\mathrm{{=}}\frac{1}{{Sin}\hspace{0.33em}{2}{u}}}\\ {}\\ {\mathrm{\Rightarrow}\frac{1}{{2}\sin\hspace{0.33em}{u}\hspace{0.33em}\cos\hspace{0.33em}{u}}}\\ {}\\ {\sin\hspace{0.33em}{u}\mathrm{{=}}\frac{\cos\hspace{0.33em}{u}\tan\hspace{0.33em}{u}}{1}\mathrm{{=}}\frac{\tan\hspace{0.33em}{u}}{\sec\hspace{0.33em}{u}}}\\ {}\\ {{Since}\hspace{0.33em}\tan{u}\mathrm{{=}}\frac{\sin\hspace{0.33em}{u}}{\cos\hspace{0.33em}{u}}}\\ {}\\ {{Also}{\mathrm{,}}\hspace{0.33em}\cos{u}\mathrm{{=}}\frac{1}{\sec\hspace{0.33em}{u}}}\\ {}\\ {\left({2}\right)\hspace{0.33em}{becomes}{\mathrm{,}}}\\ {}\\ {{I}\mathrm{{=}}\int{{\sec}^{2}u\frac{1}{\tan\hspace{0.33em}{u}\hspace{0.33em}\log\left({\tan\hspace{0.33em}{u}}\right)}dx}\hspace{0.33em}\hspace{0.33em}\mathrm{\longrightarrow}\left({3}\right)}\\ {}\\ {{Let}\hspace{0.33em}{v}\mathrm{{=}}\tan\hspace{0.33em}{u}}\\ {}\\ {{dv}\mathrm{{=}}{\sec}^{2}{u}\hspace{0.33em}{du}}\\ {}\\ {{du}\mathrm{{=}}\frac{dv}{{\sec}^{2}u}}\\ {}\\ {\left({3}\right)\hspace{0.33em}{becomes}{\mathrm{,}}}\\ {}\\ {{I}\mathrm{{=}}\frac{1}{2}\int{\frac{dv}{{v}\hspace{0.33em}\log\hspace{0.33em}{v}}}\hspace{0.33em}\hspace{0.33em}\mathrm{\longrightarrow}\left({4}\right)}\\ {}\\ {{Let}\hspace{0.33em}\log\hspace{0.33em}{v}\mathrm{{=}}\mathit{\alpha}}\\ {}\\ {{dv}\frac{1}{v}\mathrm{{=}}{d}\mathit{\alpha}}\\ {}\\ {{Substituting}\hspace{0.33em}\frac{dv}{v}\hspace{0.33em}{in}\hspace{0.33em}{eq}\left({4}\right)}\\ {}\\ {{I}\mathrm{{=}}\frac{1}{2}\int{\frac{1}{\mathit{\alpha}}}\hspace{0.33em}{d}\mathit{\alpha}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\frac{\log\hspace{0.33em}\mathit{\alpha}}{2}}\\ {}\\ {{Substitue}\hspace{0.33em}{the}\hspace{0.33em}{value}\hspace{0.33em}{of}\hspace{0.33em}\mathit{\alpha}\hspace{0.33em}{in}\hspace{0.33em}{the}\hspace{0.33em}{above}\hspace{0.33em}{equation}}\\ {}\\ {{Therefore}\hspace{0.33em}{I}\mathrm{{=}}\frac{\log\left({\log\hspace{0.33em}{v}}\right)}{2}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\frac{\log\left({\log\left({\tan\hspace{0.33em}{u}}\right)}\right)}{2}}\\ {}\\ {{Ans}\mathrm{{=}}\frac{\log\left|{\log\left({\tan\frac{x}{2}}\right)}\right|}{2}\hspace{0.33em}\mathrm{{+}}\hspace{0.33em}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Question no: 16
\begin{array}{l} {{I}\mathrm{{=}}\int{\frac{\cot\hspace{0.33em}{x}}{{3}\mathrm{{+}}{4}\log\left({\sin\hspace{0.33em}{x}}\right)}dx}}\\ {}\\ {Solution}\\ {}\\ {{Let}\hspace{0.33em}{u}\mathrm{{=}}{3}\mathrm{{+}}{4}\log\left({\sin\hspace{0.33em}{x}}\right)}\\ {}\\ {\mathrm{\Rightarrow}{du}\mathrm{{=}}{4}\mathrm{\times}\frac{1}{\sin\hspace{0.33em}{x}}\mathrm{{+}}\frac{d}{dx}\left({\sin\hspace{0.33em}{x}}\right)}\\ {}\\ {\mathrm{\Rightarrow}{du}\mathrm{{=}}{4}\hspace{0.33em}\left({\frac{\cos\hspace{0.33em}{x}}{\sin\hspace{0.33em}{x}}}\right){dx}}\\ {}\\ {\mathrm{\Rightarrow}{du}\mathrm{{=}}{4}\hspace{0.33em}\cot{x}\hspace{0.33em}{dx}}\\ {}\\ {Therefore\mathrm{,}}\\ {}\\ {{I}\mathrm{{=}}\frac{1}{4}\int{\frac{du}{u}}}\\ {}\\ {\mathrm{{=}}\frac{1}{4}\log\hspace{0.33em}{u}}\\ {}\\ {{Answer}\mathrm{{=}}\frac{1}{4}\log\left|{{3}\mathrm{{+}}{4}\log\left({\sin\hspace{0.33em}{x}}\right)}\right|\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Question no: 17
\begin{array}{l} {{I}\mathrm{{=}}\int{\frac{dx}{\sin\hspace{0.33em}{2}{x}\hspace{0.33em}\log\left({\tan\hspace{0.33em}{x}}\right)}}}\\ {}\\ {Solution}\\ {}\\ {\sin\hspace{0.33em}{2}{x}\mathrm{{=}}{2}\sin{x}\hspace{0.33em}\cos{x}}\\ {}\\ {\sin{x}\mathrm{\Rightarrow}\tan{x}\hspace{0.33em}\cos{x}\mathrm{{=}}\frac{\tan\hspace{0.33em}{x}}{\sec\hspace{0.33em}{x}}}\\ {}\\ {\cos\hspace{0.33em}{x}\mathrm{{=}}\frac{1}{\sec\hspace{0.33em}{x}}}\\ {}\\ {{Therefore}\hspace{0.33em}\sin\hspace{0.33em}{2}{x}\mathrm{{=}}\frac{{2}\hspace{0.33em}\tan{x}}{{\sec}^{2}x}}\\ {}\\ {{I}\mathrm{{=}}\frac{1}{2}\int{\frac{{\sec}^{2}x}{\tan{x}\hspace{0.33em}\log\left({\tan\hspace{0.33em}{x}}\right)}dx}}\\ {}\\ {{Let}\hspace{0.33em}{u}\mathrm{{=}}\tan\hspace{0.33em}{x}}\\ {}\\ {{du}\mathrm{{=}}{\sec}^{2}{x}\hspace{0.33em}{dx}}\\ {}\\ {\mathrm{\Rightarrow}{dx}\mathrm{{=}}\frac{du}{{\sec}^{2}x}}\\ {}\\ {Therefore\mathrm{,}}\\ {}\\ {{I}\mathrm{{=}}\frac{1}{2}\int{\frac{du}{{u}\hspace{0.33em}\log{u}}}}\\ {}\\ {{Let}\hspace{0.33em}{v}\mathrm{{=}}\log\hspace{0.33em}{u}}\\ {}\\ {\mathrm{\Rightarrow}{dv}\mathrm{{=}}\frac{du}{u}}\\ {}\\ {\mathrm{\Rightarrow}{du}\mathrm{{=}}{u}\hspace{0.33em}{dv}}\\ {}\\ {{Therefore}{\mathrm{,}}\hspace{0.33em}{I}\mathrm{{=}}\frac{1}{2}\int{\frac{dv}{v}}}\\ {}\\ {\mathrm{\Rightarrow}\frac{1}{2}\log{v}\mathrm{{=}}\frac{1}{2}\log\left|{\log\hspace{0.33em}{u}}\right|}\\ {}\\ {{Answer}\mathrm{{=}}\frac{1}{2}\log\left|{\log\left({\tan\hspace{0.33em}{x}}\right)}\right|\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Integral calculus
Question no: 18
\begin{array}{l} {{I}\mathrm{{=}}\int{\frac{{e}^{x}\mathrm{{+}}{1}}{{e}^{x}\mathrm{{-}}{1}}\hspace{0.33em}}{dx}}\\ {}\\ {{I}\mathrm{{=}}\int{\frac{{e}^{x}}{{e}^{x}\mathrm{{-}}{1}}}\hspace{0.33em}{dx}\hspace{0.33em}\mathrm{{+}}\int{\frac{1}{{e}^{x}\mathrm{{-}}{1}}\hspace{0.33em}{dx}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{\longrightarrow}\left({1}\right)}\\ {}\\ {{Consider}\hspace{0.33em}\int{\frac{{e}^{x}}{{e}^{x}\mathrm{{-}}{1}}}\hspace{0.33em}{dx}}\\ {}\\ {{Let}\hspace{0.33em}\hspace{0.33em}{t}\mathrm{{=}}{e}^{x}\mathrm{{-}}{1}}\\ {}\\ {\mathrm{\Rightarrow}{dt}\mathrm{{=}}{e}^{x}{dx}}\\ {}\\ {{Therefore}{\mathrm{,}}\hspace{0.33em}\int{\frac{{e}^{x}}{{e}^{x}\mathrm{{-}}{1}}}\hspace{0.33em}{dx}\hspace{0.33em}\mathrm{{=}}\hspace{0.33em}\int{\frac{dt}{t}\hspace{0.33em}\mathrm{{=}}\hspace{0.33em}\log\hspace{0.33em}{t}}}\\ {}\\ {\int{\frac{{e}^{x}}{{e}^{x}\mathrm{{-}}{1}}}\hspace{0.33em}{dx}\mathrm{{=}}\hspace{0.33em}\log\left({{e}^{x}\mathrm{{-}}{1}}\right)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{\longrightarrow}\left({2}\right)}\\ {}\\ {{Now}\hspace{0.33em}{consider}{\mathrm{,}}\hspace{0.33em}\int{\frac{1}{{e}^{x}\mathrm{{-}}{1}}}\hspace{0.33em}{dx}}\\ {}\\ {{Write}\hspace{0.33em}{the}\hspace{0.33em}{numerator}\hspace{0.33em}{as}\hspace{0.33em}{1}\mathrm{{+}}{e}^{x}\mathrm{{-}}{e}^{x}}\\ {}\\ {\int{\frac{1}{{e}^{x}\mathrm{{-}}{1}}}{dx}\mathrm{{=}}\int{\frac{{1}\mathrm{{-}}{e}^{x}}{{e}^{x}\mathrm{{-}}{1}}}\hspace{0.33em}{dx}\mathrm{{+}}\int{\frac{{e}^{x}}{{e}^{x}\mathrm{{-}}{1}}}\hspace{0.33em}{dx}\hspace{0.33em}}\\ {}\\ {\mathrm{{=}}\int{\frac{\mathrm{{-}}\left({{e}^{x}\mathrm{{-}}{1}}\right)}{{e}^{x}\mathrm{{-}}{1}}\hspace{0.33em}}{dx}\hspace{0.33em}\mathrm{{+}}\int{\frac{dx}{{e}^{x}\mathrm{{-}}{1}}}}\\ {}\\ {\mathrm{\Rightarrow}\mathrm{{-}}\int{{dx}\hspace{0.33em}\mathrm{{+}}\int{\frac{{e}^{x}}{{e}^{x}\mathrm{{-}}{1}}}}\hspace{0.33em}{dx}\hspace{0.33em}\mathrm{{=}}\mathrm{{-}}{x}\mathrm{{+}}\log\left({{e}^{x}\mathrm{{-}}{1}}\right)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{\longrightarrow}\left({3}\right)}\\ {}\\ {{Substitute}\hspace{0.33em}{equation}\left({2}\right){\mathrm{,}}\hspace{0.33em}\left({3}\right)\hspace{0.33em}{in}\hspace{0.33em}{eq}\left({1}\right)}\\ {}\\ {\int{\frac{{e}^{x}\mathrm{{+}}{1}}{{e}^{x}\mathrm{{-}}{1}}}\hspace{0.33em}{dx}\hspace{0.33em}\mathrm{{=}}{2}\hspace{0.33em}\log\left({{e}^{x}\mathrm{{-}}{1}}\right)\hspace{0.33em}\mathrm{{-}}{x}\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Question no: 19
\begin{array}{l} {{I}\mathrm{{=}}\int{{\sec}^{4}{x}\hspace{0.33em}\tan{x}}}{dx}\\ {}\\ {Solution}\\ {}\\ {{Let}\hspace{0.33em}{t}\mathrm{{=}}{\sec}^{4}{x}}\\ {}\\ {\mathrm{\Rightarrow}{dt}\mathrm{{=}}{4}{\sec}^{4}{x}\hspace{0.33em}\tan{x}\hspace{0.33em}{dx}}\\ {}\\ {{Therefore}\hspace{0.33em}{I}\mathrm{{=}}\frac{1}{4}\int{dt}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\frac{t}{4}}\\ {}\\ {{Answer}\mathrm{{=}}\frac{{\sec}^{4}x}{4}\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Integral calculus
Question no: 20
\begin{array}{l} {{I}\mathrm{{=}}\int{{e}^{x}\hspace{0.33em}\sin\left({{e}^{x}}\right)}\hspace{0.33em}{dx}}\\ {}\\ {Solution}\\ {}\\ {{Let}\hspace{0.33em}{e}^{x}\mathrm{{=}}{t}}\\ {}\\ {\mathrm{\Rightarrow}{e}^{x}\hspace{0.33em}{dx}\mathrm{{=}}{dt}}\\ {}\\ {{Therefore}{\mathrm{,}}\hspace{0.33em}{I}\mathrm{{=}}\int{\sin\hspace{0.33em}{t}\hspace{0.33em}{dt}}}\\ {}\\ {{I}\mathrm{{=}}\left({\mathrm{{-}}\cos\hspace{0.33em}{t}}\right)}\\ {}\\ {{Answer}\mathrm{{=}}\mathrm{{-}}\cos\left({{e}^{x}}\right)\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Question no: 21
Question no: 22
\begin{array}{l} {{I}\mathrm{{=}}\int{\frac{{\sec}^{2}x}{\sqrt{\tan\hspace{0.33em}{x}}}}\hspace{0.33em}{dx}}\\ {}\\ {Solution}\\ {}\\ {{Let}\hspace{0.33em}{t}\mathrm{{=}}\tan\hspace{0.33em}{x}}\\ {}\\ {\mathrm{\Rightarrow}{dt}\mathrm{{=}}{\sec}^{2}{x}\hspace{0.33em}{dx}}\\ {}\\ {Therefore\mathrm{,}}\\ {}\\ {{I}\mathrm{{=}}\int{\frac{dt}{\sqrt{t}}}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}{2}\sqrt{t}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}{2}\sqrt{\tan\hspace{0.33em}{x}}}\\ {}\\ {{Answer}\mathrm{{=}}{2}\sqrt{\tan\hspace{0.33em}{x}}\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Question no: 23
\begin{array}{l} {{I}\mathrm{{=}}\int{\sqrt{{\mathrm{25}}\mathrm{{-}}{9}{x}^{2}}}\hspace{0.33em}{dx}}\\ {}\\ {Solution}\\ {}\\ {{W}{\mathrm{.}}{k}{\mathrm{.}}{t}{\mathrm{,}}\hspace{0.33em}\int{\sqrt{{a}^{2}\mathrm{{-}}{x}^{2}}}\hspace{0.33em}{dx}\mathrm{{=}}\frac{1}{2}\left[{{x}\sqrt{{a}^{2}\mathrm{{-}}{x}^{2}}\mathrm{{+}}{a}^{2}{\sin}^{\mathrm{{-}}{1}}\frac{x}{a}}\right]}\\ {}\\ {{Similarly}{\mathrm{,}}\hspace{0.33em}\int{\sqrt{{\mathrm{25}}\mathrm{{-}}{9}{x}^{2}}}{dx}\mathrm{{=}}\int{\sqrt{{5}^{2}\mathrm{{-}}{\left({3x}\right)}^{2}}}\hspace{0.33em}{dx}}\\ {}\\ {{I}\mathrm{{=}}\frac{1}{2}\left[{\frac{x\sqrt{{\mathrm{25}}\mathrm{{-}}{9}{x}^{2}}}{3}\mathrm{{+}}\frac{\mathrm{25}}{3}{\sin}^{\mathrm{{-}}{1}}\left({\frac{3x}{5}}\right)}\right]\mathrm{{+}}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Question no: 24
\begin{array}{l} {{I}\mathrm{{=}}\int{\sqrt{{2}{ax}\mathrm{{-}}{x}^{2}}}\hspace{0.33em}{dx}}\\ {}\\ {{Solution}\hspace{0.33em}}\\ {}\\ {{We}\hspace{0.33em}{know}\hspace{0.33em}{that}{\mathrm{,}}\hspace{0.33em}{\left({{x}\mathrm{{-}}{a}}\right)}^{2}\mathrm{{=}}{a}^{2}\mathrm{{-}}{2}{ax}\mathrm{{+}}{x}^{2}}\\ {}\\ {\mathrm{\Rightarrow}{x}^{2}\mathrm{{=}}{\mathrm{(}}{x}\mathrm{{-}}{a}{\mathrm{)}}^{2}\mathrm{{-}}{a}^{2}\mathrm{{+}}{2}{ax}}\\ {}\\ {\mathrm{\Rightarrow}{2}{ax}\mathrm{{=}}{x}^{2}\mathrm{{-}}{\left({{x}\mathrm{{-}}{a}}\right)}^{2}\mathrm{{+}}{a}^{2}}\\ {}\\ {{Therefore}\hspace{0.33em}{I}\hspace{0.33em}{becomes}{\mathrm{,}}}\\ {}\\ {{I}\mathrm{{=}}\int{\mathrm{[(}{x}^{2}}\mathrm{{-}}{\left({{x}\mathrm{{-}}{a}}\right)}^{2}\mathrm{{+}}{a}^{2}{\mathrm{)}}\mathrm{{-}}{x}^{2}{\mathrm{]}}\hspace{0.33em}{dx}}\\ {}\\ {\hspace{0.33em}\hspace{0.33em}\mathrm{{=}}\int{\left({{a}^{2}\mathrm{{-}}{\left({{x}\mathrm{{-}}{a}}\right)}^{2}}\right)}\hspace{0.33em}{dx}}\\ {}\\ {{Let}\hspace{0.33em}{t}\mathrm{{=}}\hspace{0.33em}{x}\mathrm{{-}}{a}}\\ {}\\ {\mathrm{\Rightarrow}{dt}\mathrm{{=}}{dx}}\\ {}\\ {{Therefore}{\mathrm{,}}\hspace{0.33em}{I}\mathrm{{=}}\int{\left({{a}^{2}\mathrm{{-}}{t}^{2}}\right)}\hspace{0.33em}{dt}}\\ {}\\ {{I}\mathrm{{=}}\frac{1}{2}\left[{{t}\sqrt{{a}^{2}\mathrm{{-}}{t}^{2}}\mathrm{{+}}{a}^{2}{\sin}^{\mathrm{{-}}{1}}\left({\frac{t}{a}}\right)}\right]}\\ {}\\ {{Answer}\mathrm{{=}}\frac{1}{2}\left[{\left({{x}\mathrm{{-}}{a}}\right)\sqrt{{2}{ax}\mathrm{{-}}{x}^{2}}\mathrm{{+}}{a}^{2}{\sin}^{\mathrm{{-}}{1}}\left({\frac{{x}\mathrm{{-}}{a}}{a}}\right)}\right]\hspace{0.33em}\mathrm{{+}}\hspace{0.33em}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Question no: 25
\begin{array}{l} {{I}\mathrm{{=}}\int{\sqrt{{3}{x}^{2}\mathrm{{+}}{4}}\hspace{0.33em}{dx}}}\\ {}\\ {{Solution}\hspace{0.33em}}\\ {}\\ {{I}\mathrm{{=}}\int{\sqrt{{\left({\sqrt{3}x}\right)}^{2}\mathrm{{+}}{2}^{2}}}\hspace{0.33em}{dx}}\\ {}\\ {\int{\sqrt{{x}^{2}\mathrm{{+}}{a}^{2}}}\hspace{0.33em}{dx}\hspace{0.33em}\mathrm{{=}}\frac{1}{2}\left[{\left({x\sqrt{{a}^{2}\mathrm{{+}}{x}^{2}}}\right)\mathrm{{+}}{a}^{2}\log\left|{{x}\mathrm{{+}}\sqrt{{a}^{2}\mathrm{{+}}{x}^{2}}}\right|}\right]}\\ {}\\ {{Therefore}{\mathrm{,}}\hspace{0.33em}\int{\sqrt{{\left({\sqrt{3}x}\right)}^{2}\mathrm{{+}}{2}^{2}}}\hspace{0.33em}{dx}\hspace{0.33em}\mathrm{{=}}\frac{1}{2}\left[{\frac{\sqrt{3}x\sqrt{{3}{x}^{2}\mathrm{{+}}{4}}}{\sqrt{3}}\mathrm{{+}}\frac{4}{\sqrt{3}}\log\left|{\frac{\sqrt{3}{x}\mathrm{{+}}\sqrt{{3}{x}^{2}\mathrm{{+}}{4}}}{1}}\right|}\right]}\\ {}\\ {{Answer}\mathrm{{=}}\frac{x\sqrt{{3}{x}^{2}\mathrm{{+}}{4}}}{2}\mathrm{{+}}\frac{2}{\sqrt{3}}\log\left|{\sqrt{3}{x}\mathrm{{+}}\sqrt{{3}{x}^{2}\mathrm{{+}}{4}}}\right|\hspace{0.33em}\mathrm{{+}}\hspace{0.33em}{C}}\\ {}\\ {{Where}\hspace{0.33em}{C}\hspace{0.33em}{is}\hspace{0.33em}{the}\hspace{0.33em}{int}{egration}\hspace{0.33em}{cons}\tan{t}{\mathrm{.}}} \end{array}Question no: 26
\[ I=\int \sqrt{9x^2+1}\,dx \] \[ \text{Solution} \] \[ =\int \sqrt{(3x)^2+1^2}\,dx \] \[ \text{Using the formula:} \] \[ \int \sqrt{x^2+a^2}\,dx = \frac{1}{2} \left[ x\sqrt{x^2+a^2} + a^2\log\left|x+\sqrt{x^2+a^2}\right| \right] +C \] \[ \therefore \] \[ \int \sqrt{(3x)^2+1^2}\,dx = \frac{1}{2} \left[ x\sqrt{9x^2+1} + \frac{1}{3} \log\left|3x+\sqrt{9x^2+1}\right| \right] +C \] \[ \text{Answer} \] \[ I= \frac{x\sqrt{9x^2+1}}{2} + \frac{1}{6} \log\left|3x+\sqrt{9x^2+1}\right| + C \]Question no: 27
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