Chapter 7

Use integration by parts to evaluate the integrals. $$ \int x \cos x d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int 2 x \sqrt{x^{2}+3} d x, \text { with } u=x^{2}+3 $$

Use the midpoint rule to approximate each integral with the specified value of \(n\). \(\int_{1}^{2} x^{2} d x, n=4\)

In Problems 1-5, find the linear approximation of \(f(x)\) at \(x=0\). $$ f(x)=e^{2 x} $$

All the integrals in Problems \(1-16\) are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{0}^{\infty} 3 e^{-6 x} d x $$

In Problems \(1-4\), use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{2 x^{2}+5 x-1}{x+2} $$

Use integration by parts to evaluate the integrals. $$ \int 3 x \cos x d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int 3 x^{2} \sqrt{x^{3}+1} d x, \text { with } u=x^{3}+1 $$

Use the midpoint rule to approximate each integral with the specified value of \(n\). \(\int_{-1}^{0}(x+1)^{3} d x, n=5\)

In Problems 1-5, find the linear approximation of \(f(x)\) at \(x=0\). $$ f(x)=\sin (3 x) $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{0}^{\infty} x e^{-x} d x $$

In Problems , use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=-\frac{x^{2}-4 x-1}{x-1} $$

Use integration by parts to evaluate the integrals. $$ \int 2 x \cos (3 x-1) d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int 3 x\left(1-x^{2}\right)^{1 / 4} d x, \text { with } u=1-x^{2} $$

Use the midpoint rule to approximate each integral with the specified value of \(n\). \(\int_{0}^{1} e^{-x} d x, n=3\)

In Problems 1-5, find the linear approximation of \(f(x)\) at \(x=0\). $$ f(x)=\frac{1}{1-x} $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{0}^{\infty} \frac{2}{1+x^{2}} d x $$

In Problems , use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function.$$ f(x)=\frac{3 x^{3}+5 x-2 x^{2}-2}{x^{2}+1} $$

Use integration by parts to evaluate the integrals. $$ \int 3 x \cos (4-x) d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int 4 x^{3}\left(4+x^{4}\right)^{1 / 3} d x, \text { with } u=4+x^{4} $$

Use the midpoint rule to approximate each integral with the specified value of \(n\). \(\int_{0}^{\pi / 2} \sin x d x, n=4\)

In Problems 1-5, find the linear approximation of \(f(x)\) at \(x=0\). $$ f(x)=x^{4} $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{e}^{\infty} \frac{d x}{x(\ln x)^{2}} $$

In Problems , use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}-3 x^{2}-15}{x^{2}+x+3} $$

Use integration by parts to evaluate the integrals. $$ \int 2 x \sin (x-1) d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int 5 \cos (3 x) d x, \text { with } u=3 x $$

Use the midpoint rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. \(\int_{2}^{4} \frac{1}{x} d x, n=4\)

In Problems 1-5, find the linear approximation of \(f(x)\) at \(x=0\). $$ f(x)=\ln \left(2+x^{2}\right) $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{1}^{\infty} \frac{1}{x^{3 / 2}} d x $$

In Problems 5-8, write out the partial-fraction decomposition of the function \(f(x) .\) $$ f(x)=\frac{2 x-3}{x(x+1)} $$

Use integration by parts to evaluate the integrals. $$ \int x \sin (1-2 x) d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int 5 \sin (1-2 x) d x, \text { with } u=1-2 x $$

Use the midpoint rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. \(\int_{-1}^{1}\left(e^{2 x}-1\right) d x, n=4\)

In Problems \(6-10\), compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions. $$ f(x)=\frac{1}{1+x}, n=4 $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{-\infty}^{-1} \frac{1}{1+x^{2}} d x $$

In Problems , write out the partial-fraction decomposition of the function \(f(x) .\) $$ f(x)=-\frac{x+1}{(2 x+1)(x-1)} $$

Use integration by parts to evaluate the integrals. $$ \int x e^{x} d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int 7 x^{2} \sin \left(4 x^{3}\right) d x, \text { with } u=4 x^{3} $$

Use the midpoint rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. \(\int_{0}^{4} \sqrt{x} d x, n=4\)

In Problems \(6-10\), compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions. $$ f(x)=\cos x, n=5 $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{-\infty}^{\infty} e^{-|x|} d x $$

In Problems , write out the partial-fraction decomposition of the function \(f(x) .\) $$ f(x)=\frac{4 x^{2}-14 x-6}{x(x-3)(x+1)} $$

Use integration by parts to evaluate the integrals. $$ \int 3 x e^{-x / 2} d x $$

Evaluate the indefinite integral by making the given substitution. $$ \int x \cos \left(x^{2}-1\right) d x, \text { with } u=x^{2}-1 $$

Use the midpoint rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. \(\int_{2}^{4} \frac{2}{\sqrt{x}} d x, n=5\)

In Problems \(6-10\), compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions. $$ f(x)=e^{3 x}, n=3 $$

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{-\infty}^{\infty} x e^{-x^{2} / 2} d x $$

In Problems , write out the partial-fraction decomposition of the function $f(x) .$$$ f(x)=\frac{16 x-6}{(2 x-5)(3 x+1)} $$

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

Evaluate the indefinite integral by making the given substitution. $$ \int e^{2 x+3} d x, \text { with } u=2 x+3 $$