91Ó°ÊÓ

Use the theoretical error bound to determine how large n should be. [Hint: In each case, find the second derivative of the integrand you can use a graphing calculator to find an upper bound on \(\left.\left|f^{\prime \prime}(x)\right| .\right]\) How large should \(n\) be so that the trapezoidal rule approximation of $$ \int_{0}^{1} e^{-x} d x $$ is accurate to within \(10^{-5}\) ?

Evaluating the integral \(\int \arccos x d x\) requires two steps. (a) Write $$ \arccos x=1 \cdot \arccos x $$ and integrate by parts once to show that $$ \int \arccos x d x=x \arccos x+\int \frac{x}{\sqrt{1-x^{2}}} d x $$ (b) Use substitution to compute $$ \int \frac{x}{\sqrt{1-x^{2}}} d x $$ and combine your result in (a) with (7.12) to complete the computation of \(\int \arccos x d x\).

Determine the constant \(c\) so that $$ \int_{-\infty}^{\infty} \frac{c}{1+x^{2}} d x=1 $$

Use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{1}{(x-1)(x+2)} d x $$

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int \frac{d x}{(x+3) \ln (x+3)} $$

In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int x^{3} \sqrt{1+x^{2}} d x $$

Use the theoretical error bound to determine how large n should be. [Hint: In each case, find the second derivative of the integrand you can use a graphing calculator to find an upper bound on \(\left.\left|f^{\prime \prime}(x)\right| .\right]\) How large should \(n\) be so that the trapezoidal rule approximation of $$ \int_{1}^{2} \frac{e^{t}}{t} d t $$ is accurate to within \(10^{-4}\) ?

In this problem, we investigate the integral \(\int_{1}^{\infty} \frac{1}{x^{p}} d x\) for \(01\), set \(A(z)=\int_{1}^{z} \frac{1}{x^{p}} d x\) and show that $$A(z)=\frac{1}{1-p}\left(z^{-p+1}-1\right) \quad \text { for } p \neq 1$$ and $$A(z)=\ln z \quad \text { for } p=1$$ (b) Use your results in (a) to show that, for \(0

1\), $$ \lim _{z \rightarrow \infty} A(z)=\frac{1}{p-1} $$

Use partial-fraction decomposition to evaluate the integrals. $$ \int \frac{x^{2}-2 x-2}{x(x+2)} d x $$

(a) Use integration by parts to show that, for \(x>0\), $$ \int \frac{1}{x} \ln x d x=(\ln x)^{2}-\int \frac{1}{x} \ln x d x $$ (b) Use your result in (a) to evaluate $$ \int \frac{1}{x} \ln x d x $$