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

1

(a)

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

Evaluate the integral: \(\int {(ax\ +\ b)^3}dx\)

1

(a)

\({(ax +b)^4\over 4a}+c\)

\(\int cos^2x\ cosec^2x\ dx.\)

1

(a)

\(= -{1\over3}cot^3\ x+c\)

\(\int {x\over \sqrt {x+2}}dx.\)

1

(a)

\(={2\over3}(x+2)^{3/2}-4\sqrt{x+2}+c\)

Given \(\int { { e }^{ x }(tanx+1)secxdx={ e }^{ x }f(x)+c } \). Write f(x) satisfying the above.

1

(a)

\(I=\int { { e }^{ x }(secxtanx+secx)dx } \)
\(={ e }^{ x }secx+C\)
\([\because \int { { e }^{ x }\left\{ f(x)+f'(x) \right\} } dx={ e }^{ x }f(x)+C\)
 \(\therefore \quad f(x)=secx\)

Evaluate : \(\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 2 } } } \)

1

(a)

\(\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 2 } } } =\left[ { tan }^{ -1 }x \right] ^{ 1 }_{ 0 }\)
\(={ tan }^{ -1 }1-{ tan }^{ -1 }0\)
\(=\frac { \pi }{ 4 } -0=\frac { \pi }{ 4 } \)

\(\int { sin2xcos3xdx } \)

1

(a)

\(\int { { e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ x^{ 2 } } \right) } dx\)

1

(a)

Evaluate : \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { dx }{ 1+\sqrt { tanx } } } \)

1

(a)

Find: \(\int { \frac { 1 }{ (x+1)(x+2) } } dx\)

1

(a)

Evaluate:\(\int _{ 0 }^{ \pi /2 }{ \frac { \sin ^{ 4 }{ x } }{ \sin ^{ 4 }{ x } +\cos ^{ 4 }{ x } } } dx.\)

2

(a)

Find the integral: \(\int { { (ax }^{ 2 }+bx+c } )dx\)

2

(a)

Evaluate the definite integral in  \(\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 3 } }{ \frac { \sin { x } +\cos { x } }{ \sqrt { \sin { 2x } } } } dx.\)

2

(a)

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

2

(a)

Evaluate : \(\int { \frac { { \left( \log { x } \right) }^{ 2 } }{ x } } dx\)

2

(a)

\(\int { \frac { (x+3){ e }^{ x } }{ { (x+5) }^{ 3 } } } dx\)

6

(a)

Evaluate : \(\int { \frac { 3x+1 }{ (x+1)^{ 2 }(x+3) } } dx\)

6

(a)

Evaluate : \(\int _{ 2 }^{ 5 }{ \left[ \left| x-2 \right| +\left| x-3 \right| +\left| x-5 \right| \right] } dx\)

6

(a)

Evaluate ; \(\int { (5x-1)\sqrt { 6+5x-2x^{ 2 } } } dx\)

6

(a)

Evaluate : \(\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 3 } }{ \frac { dx }{ 1+\sqrt { \cot { x } } } } \)

6

(a)