If f'(x) = a sin x + b cos x and f'(0) = 4, f(0) = 3, f\(\left( \frac { \pi }{ 2 } \right) \) = 5, find f(x).

Evaluate \(\int { \frac { { x }^{ 7 } }{ { x }^{ 5 }+1 } } dx\)

Evaluate ഽ x³ sin (x⁴) dx

Evaluate \(\int { \frac { 1 }{ { 3x }^{ 2 }+13x-10 } } dx\)

Evaluate ഽx. log (1 + x) dx

Evaluate ഽ sin (log x) + cos (log x) dx

Evaluate \(\int { \frac { \left( { x }^{ 2 }+1 \right) dx }{ { \left( x-1 \right) }^{ 2 }\left( x+3 \right) } } \)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ sin } 3xsin\ 2x\ dx\)

Prove that \(\int _{ a }^{ b }{ \frac { f\left( x \right) }{ f\left( x \right) +f(a+b-x) } } dx=\frac { b-a }{ 2 } \)

Using integrals as limit of sums, evaluate \(\int _{ 2 }^{ 4 }{ (2x-1) } dx\)

Evaluate \(\int\left[\frac{2+x+x^{2}}{x^{2}(2+x)}+\frac{2 x-1}{(x+1)^{2}}\right] d x\)

\(\int \frac{x^{3}}{(x+1)^{2}} d x\)

\(\int \frac{1}{3 x^{2}+13 x-10} d x\)

\(\int \frac{x^{3}+x}{x^{4}-9} d x\)

\(\int x^{3} e^{x} d x\)