Evaluate the integral: \(\int { {x^2\over1+x^3}dx. } \)

1

(a)

\(\int \frac{x^{2}}{1+x^{3}} d x =\frac{1}{3} \int \frac{1}{t} d t=\frac{1}{3} \log |t|+C \)
\(=\frac{1}{3} \log \left|1+x^{3}\right|+C \)

\(\int {1+tan\ x\over 1-tan\ x}dx\).

1

(a)

\(\int \frac{1+\tan x}{1-\tan x} d x =\int \frac{\cos x+\sin x}{\cos x-\sin x} d x \)
\(=-\int \frac{1}{t} d t \)
\(=-\log |t|+C \)
\(=-\log |\cos x-\sin x|+C\)

\(\int {dx\over x\ cos^2 (1+log\ x)}\).

1

(a)

\(\int \frac{d x}{x \cos ^{2}(1+\log x)} =\int \sec ^{2} t d t \)
\(=\tan t+C \)
= tan |1 + log x| + c

\(\int {x^2-1\over x^2+1}dx.\)

1

(a)

= x-2 tan^-1x + c

\(\int {e^x\over 1+e^x}dx.\)

1

(a)

\(=\log { \left| 1+{ e }^{ x } \right| } +c\)

\(\int cos\ 3x\ cos\ x\ dx.\)

1

(a)

\(={sin\ 4x\over8}+{sin\ 2x\over4}+c\)

Find: \(\int { \frac { { sin }^{ 2 }x-{ cos }^{ 2 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } } dx\)

1

(a)

\(\int \frac{\sin ^{2} x-\cos ^{2} x}{\sin x \cos x} d x=\int(\tan x-\cot x) d x=\log |\sec x|-\log |\sin x|+C\)

Evaluate : \(\int _{ 2 }^{ 4 }{ \frac { x }{ { x }^{ 2 }+1 } } dx\)

1

(a)

Put,  \(1+{ x }^{ 2 }=t\)
\(\Rightarrow 2xdx=dt\)
\(\Rightarrow xdx=\frac { dt }{ 2 } \)
\(\therefore \int _{ 2 }^{ 4 }{ \frac { x }{ { x }^{ 2 }+1 } } dx=\int _{ 2 }^{ 4 }{ \frac { dt }{ 2t } =\frac { 1 }{ 2 } \left[ logt \right] ^{ 4 }_{ 2 } } \)
\(\frac { 1 }{ 2 } \left[ log(1+{ x }^{ 2 }) \right] ^{ 4 }_{ 2 }=\frac { 1 }{ 2 } log\frac { 17 }{ 5 } \)

Evaluate:\(\int _{ 0 }^{ x }{ t } sintdt\) then write the value of 

1

(a)

\(\\ \int _{ 0 }^{ x }{ t } sintdt=f(x),\)
\(\therefore \ f'(x)=xsinx\)

\(\int _{ 0 }^{ a }{ 3{ x }^{ 2 } } dx=8\) , then write the value of a.

1

(a)

\(\left[ \frac { 3x^{ 3 } }{ 3 } \right] ^{ 1 }_{ 0 }=8\)
\(\Rightarrow \left[ x^{ 3 } \right] ^{ a }_{ 0 }=8\)
\(\Rightarrow { a }^{ 3 }=8\)
\(\Rightarrow a=2\)

\(\int { \frac { dx }{ 1+sinx } } \)

2

(a)

\(\int { \frac { dx }{ 1+{ e }^{ x } } } \)

2

(a)

Evaluate : \(\int _{ 2 }^{ 8 }{ \left| x-5 \right| } dx\)

2

(a)

Evaluate : \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\ tanxdx } \)

2

(a)

Evaluate \(\int _{ 0 }^{ 1 }{ 5x\sqrt { 5-{ x }^{ 2 } } } dx.\)

3

(a)

Evaluate:\(\int _{ -\pi /2 }^{ -\pi /2 }{ \sin ^{ 5 }{ xdx } } .\)

3

(a)

Find the following integrals: 
\((i) \int { (\sin { x } } +\cos { x } )dx\)
\((ii) \int { co\sec { x } ( } co\sec { x } +\cot { x } )dx\)
\((iii) \int { \frac { 1-\sin { x } }{ \cos ^{ 2 }{ x } } } dx\)

3

(a)

Find: \(\int { \sqrt { { x }^{ 2 }+2x+5 } } dx.\)

3

(a)

Evaluate the integral: \(\int {x^2+4\over x^4+16}dx.\)

5

(a)

Evaluate the integral: \(\int x^2 log (1+x) dx.\)

5

(a)

Find : \(\int { \frac { 1 }{ { cos }^{ 4 }x+{ sin }^{ 4 }x } } dx\)

5

(a)

Evaluate : \(\int _{ 0 }^{ \pi }{ \frac { x }{ { a }^{ 2 }\cos ^{ 2 }{ x } +{ b }^{ 2 }\sin ^{ 2 }{ x } } } dx\)

5

(a)