91Ó°ÊÓ

Use integration by parts to evaluate the integrals. $$ \int_{1}^{2} \ln x d x $$

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{4} \frac{1}{x^{1 / 4}} d x $$

Use substitution to evaluate the indefinite integrals. $$ \int\left(x^{2}-2 x\right)\left(x^{3}-3 x^{2}+3\right)^{2 / 3} d x $$

In Problems 19-23, compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=\ln x, a=1, n=3 ; x=2 $$

Use integration by parts to evaluate the integrals. $$ \int_{1}^{e} \ln x^{2} d x $$

In Problems , use partial-fraction decompositon to evaluate each integral. $$ \int \frac{x^{3}-3 x^{2}+x-6}{\left(x^{2}+2\right)\left(x^{2}+1\right)} d x $$

In Problems , use partial-fraction decompositon to evaluate each integral. $$ \int \frac{2 x^{2}-3 x+2}{\left(x^{2}+1\right)^{2}} d x $$

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{0}^{2} \frac{1}{(x-1)^{1 / 3}} d x $$

How large should \(n\) be so that the trapezoidal rule approximation of $$ \int_{0}^{1} e^{-x} d x $$ is accurate to within \(10^{-5}\) ?

In Problems 19-23, compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=\cos x, a=\frac{\pi}{6}, n=3 ; x=\frac{\pi}{7} $$