91Ó°ÊÓ

In Problems 1-30, use integration by parts to evaluate each integral. $$ \int_{0}^{\pi / 3} x \sin x d x $$

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \sqrt{x+2} d x $$

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{0}^{1} x^{2} d x\) (a) \(n=10\) (b) \(n=20\)

(a) Find the Taylor polynomial of degree 3 about \(x=0\) for \(f(x)=\sin x\). (b) Use your result in (a) to explain why $$ \lim _{x \rightarrow 0} \frac{\sin x}{x}=1 $$

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{\infty} \frac{1}{x+1} d x $$

Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{2 x-3}{x(x+1)} $$

Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int(x-1)^{2} e^{2 x} d x $$

(a) Find the Taylor polynomial of degree 2 about \(x=0\) for \(f(x)=\cos x\) (b) Use your result in (a) to explain why $$ \lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0 $$

Use a spreadsheet to approximate each of the following integrals using the midpoint rule with each of the specified values of \(n .\) \(\int_{-1}^{1} \sqrt{x+1} d x\) (a) \(n=10\) (b) \(n=30\).

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int(4+x)^{1 / 7} d x $$