91Ó°ÊÓ

Solve the initial value problems in Exercises \(67-70\) for \(x\) as a function of \(t .\) $$ \left(t^{2}-3 t+2\right) \frac{d x}{d t}=1 \quad(t>2), \quad x(3)=0 $$

Use any method to evaluate the integrals. \(\int x \sin ^{2} x d x\)

Use integration by parts to establish the reduction formula. $$ \int x^{n} \cos x d x=x^{n} \sin x-n \int x^{n-1} \sin x d x $$

Solve the initial value problems in Exercises \(67-70\) for \(x\) as a function of \(t .\) $$ \left(3 t^{4}+4 t^{2}+1\right) \frac{d x}{d t}=2 \sqrt{3}, \quad x(1)=-\pi \sqrt{3} / 4 $$

Use any method to evaluate the integrals. \(\int x \cos ^{3} x d x\)

In Exercises \(35-68\) , use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{-\infty}^{\infty} \frac{d x}{e^{x}+e^{-x}}$$

Use integration by parts to establish the reduction formula. $$ \int x^{n} \sin x d x=-x^{n} \cos x+n \int x^{n-1} \cos x d x $$

Arc length Find the length of the curve $$y=\ln (\sin x), \quad \frac{\pi}{6} \leq x \leq \frac{\pi}{2}$$

Solve the initial value problems in Exercises \(67-70\) for \(x\) as a function of \(t .\) $$ \left(t^{2}+2 t\right) \frac{d x}{d t}=2 x+2 \quad(t, x>0), \quad x(1)=1 $$

Use integration by parts to establish the reduction formula. $$ \int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad a \neq 0 $$