For $x \neq -1$, if $\int \frac{xe^x dx}{(1+x)^2} = \frac{ae^x}{(1+x)^b} + c$, where a, b are fixed numbers and c is the integration constant, then $a + b$ is equal to

$\int \dfrac{\pi}{x^{n+1}-x} d x=$

$\displaystyle\int_{-2}^{2} \dfrac{x^2}{1+5^x}\, dx$ is equal to:

The integral $\int e^x\left(\frac{x-1}{2x^2}\right)dx$ is equal to

$\int (x^4 + x^2 + 1)d(x^2)$ is equal to: (where c is an integration constant)

$\int \frac{e^{7\log_e x} - e^{6\log_e x}}{e^{4\log_e x} - e^{3\log_e x}} dx$ is equal to:

(Here, c is an arbitrary constant)

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