Evaluate the following definite integrals: int_0^1(x e^(-x^2+2))dx

Evaluate the following integral.
$\int \frac{3x^2+\sqrt{x}}{\sqrt{x}}dx$

Use any method to evaluate the integral.
${\displaystyle \int \frac{x}{25+4x^2}dx}$

I do not understand why for the two integrals below, the result is always 0 unless $|k|=n$ , in which case the result is $\pi /2$
${\int }_0^{\pi }d\theta \cos k\theta \cos n\theta {\int }_0^{\pi }d\theta \sin k\theta \sin n\theta$
I have tried using trigonometric identities to get a general solution but I had no luck understanding the nature of the integrals. If anyone can point out any hints or patterns, it would be much appreciated.

Сalculate the integral.
${\displaystyle {\int }_0^3\frac{dx}{x^2+3}}$

Express the limits as definite integral.
${\displaystyle \underset{\Vert p\to 0\Vert }{lim}\sum _{k=1}^n\left(\frac{1}{c_k}\right)\mathrm{\Delta }{x}_k,wherePisapartitionof[1,4]}$

Find the trigonometric integral ${\displaystyle \int {\sec }^4\left(\frac{x}{2}\right)dx}$

Evaluate the following integral.
${\displaystyle \int (5+x+{\tan }^{2}x)dx}$