91Ó°ÊÓ

Compute the indefinite integrals. $$ \int x e^{-x^{2} / 2} d x $$

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=\ln x, 1 \leq x \leq e\)

Compute the indefinite integrals. $$ \int e^{x}\left(1-e^{-x}\right) d x $$

Use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{-3}^{3} \sqrt{9-x^{2}} d x $$

Compute the indefinite integrals. $$ \int \sin (2 x) d x $$

Use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{2}^{5}\left(\frac{1}{2} x-4\right) d x $$

Compute the indefinite integrals. $$ \int \sin \frac{1-x}{3} d x $$

Use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{1 / 2}^{1} \sqrt{1-x^{2}} d x $$

A cable that hangs between two poles at \(x=-M\) and \(x=M\) takes the shape of a catenary, with equation $$ y=\frac{1}{2 a}\left(e^{a x}+e^{-a x}\right) $$ where \(a\) is a positive constant. Compute the length of the cable when \(a=1\) and \(M=\ln 2\).

Use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{-2}^{2}\left(\sqrt{4-x^{2}}-2\right) d x $$