Value of the definite integral $\int_{-3}^{1} (2(t+1)^5 - 5(t+1)^3 + t + 3) dt$ is equal to

$\int_{-1}^1 \sin ^7 x \cdot \cos ^6 x \, dx = $ . . . . . . .

By using the properties of definite integrals,evaluate the integral $\int_{0}^{4}|x-1| d x$.

$\int_0^{2 \pi} \sin ^3 x \cos ^2 x \, dx = $ . . . . . . .

Let $f, f', f''$ be continuous in $[0, \ln 2]$ and $f(0) = 0, f'(0) = 3, f(\ln 2) = 6, f'(\ln 2) = 4$ and $\int_{0}^{\ln 2} e^{-2x} f(x) dx = 3$,then $\int_{0}^{\ln 2} e^{-2x} f''(x) dx$ is

If $\int_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x \, dx}{(1+e^{\sin x})(1+\sin ^4 x)} = \alpha \pi + \beta \log _e(3+2 \sqrt{2})$,where $\alpha, \beta$ are integers,then $\alpha^2+\beta^2$ equals.....................