Indefinite Integrals

If $\int {{x^5}} {e^{ - {x^2}}}dx = g\left( x \right){e^{ - {x^2}}} + c$, where c is a constant of integration, then $g$(–1) is equal to :

If $\int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}} \right) + {{f\left( x \right)} \over {{x^2} - 2x + 10}}} \right) + C$ where C is a constant of integration then :

If $\int {{{dx} \over {{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){{(1 + {x^6})}^{{1 \over 3}}} + C}$ where C is a constant of integration, then the function ƒ(x) is equal to

The integral $\int {{{3{x^{13}} + 2{x^{11}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}} \,dx$ is equal to : (where C is a constant of integration)

If   $\int {{{x + 1} \over {\sqrt {2x - 1} }}} \,dx$ = f(x) $\sqrt {2x - 1}$ + C, where C is a constant of integration, then f(x) is equal to :

If  $\int {{{\sqrt {1 - {x^2}} } \over {{x^4}}}}$ dx = A(x)${\left( {\sqrt {1 - {x^2}} } \right)^m}$ + C, for a suitable chosen integer m and a function A(x), where C is a constant of integration, then (A(x))^m equals :

If  $\int \,$x⁵.e$-$4x³ dx = ${1 \over {48}}$e$-$4x³ f(x) + C, where C is a constant of inegration, then f(x) is equal to -

If   $f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right),$ $f\left( 0 \right) = 0,$    then the value of $f(1)$ is :

For x² $\ne$ n$\pi$ + 1, n $\in$ N (the set of natural numbers), the integral $\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx}$ is equal to : (where c is a constant of integration)

If $\int {{{d\theta } \over {{{\cos }^2}\theta \left( {\tan 2\theta + \sec 2\theta } \right)}}} = \lambda \tan \theta + 2{\log _e}\left| {f\left( \theta \right)} \right| + C$ where C is a constant of integration, then the ordered pair ($\lambda$, ƒ($\theta$)) is equal to :