$\int \frac{x^2+1}{(x+1)^2} e^x \, dx$ is equal to

$\int \frac{1}{x(x^6+1)}\,dx =$ (consider $\log_e x = \log x$)

$\displaystyle\int e^{x\log_e 7} e^x\, dx$ is equal to :

[where C is an arbitrary constant]

$\int \frac{x^8}{(x^3+1)^{1/3}} dx$ is equal to

Value of $\int \frac{2}{(x-3)\sqrt{x+1}} dx$ is: (Here C is an arbitrary constant)

$\int \frac{x^2 + 4}{(x^2 + 3)(x^2 + 5)} \, dx = u \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + v \tan^{-1}\left(\frac{x}{\sqrt{5}}\right) + c$, where c is arbitrary constant, then value of $\frac{1}{u^2} + \frac{1}{v^2}$ is equal to :

$\int \frac{dx}{2\sin^2 x + 5\cos^2 x}$ is equal to