$\int_{-1}^{1} \sin^3 x \cos^2 x \, dx = $

$\int_{0}^{1} \tan ^{-1}\left(\frac{2x-1}{1+x-x^{2}}\right) dx =$

જો $I_{n}=\int_0^{\frac{\pi}{4}} \tan ^n x \, dx$ હોય,તો $I_{13}+I_{11}=$

$\int_{0}^{1} \frac{\log (1+x)}{1+x^{2}} d x$ ની કિંમત શોધો.

જો $\int \limits_0^1 \frac{1}{\left(5+2 x -2 x ^2\right)\left(1+ e ^{(2-4 x)}\right)} dx =\frac{1}{\alpha} \log _{ e }\left(\frac{\alpha+1}{\beta}\right)$ જ્યાં $\alpha, \beta > 0$,તો $\alpha^4-\beta^4$ ની કિંમત શોધો:

$\int_0^{\pi} x f(\sin x) \, dx$ ની કિંમત શોધો.