7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
19. ∫ 1/√(x² - 81) dx, x > 9

Evaluate the definite integral.

${\int }_0^{\frac{\pi }{3}}\frac{sin\theta }{1+\cos \theta }d\theta$

Find the area under the graph of $f\left(x\right)=\frac{e^{-x}}{1+e^{-x}}$ from $x=0$ to $x=4$.

The region between the curve $y=\frac{2}{\sqrt{x}}$ and the $x$-axis from $x=1$ to $x=3$ is revolved about the $x$-axis to form a solid. Find the volume of this solid. 

Solve the initial value problem given by $\frac{dy}{dx}=\frac{2}{x}+3,y\left(1\right)=4$.

Find the indefinite integral.

${\int }_{}^{}\frac{3-y^2}{2y}dy$

Evaluate the definite integral.

${\int }_2^e\frac{3+x}{x}dx$

Find the area under the graph of $f\left(t\right)=\frac{t^2{e}^t+t}{t^2}$ from $t=1$ to $t=3$.

Find the indefinite integral.

${\int }_{}^{}\frac{1}{2x+5}dx$