Chapter 7

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

Use integration by parts to evaluate the integrals. $$ \int_{0}^{\pi / 4} 2 x \cos x d x $$

Use substitution to evaluate the indefinite integrals. $$ \int(4-x)^{1 / 7} d x $$

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

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

In Problems , use partial-fraction decomposition to evaluate the integrals.$$ \int \frac{4 x^{2}-x-1}{(x+1)^{2}(x-3)} d x $$

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

Use substitution to evaluate the indefinite integrals. $$ \int(4 x-3) \sqrt{2 x^{2}-3 x+2} d x $$

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 \(x\). $$ f(x)=\sqrt{x}, a=1, n=3 ; x=2 $$

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

In Problems 19-22, use partial-fraction decompositon to evaluate each integral. $$ \int \frac{x^{2}-x^{2}+x-4}{\left(x^{2}+1\right)\left(x^{2}+4\right)} d x $$

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

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 \(x\). $$ f(x)=\ln x, a=1, n=3 ; x=2 $$

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

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

Use integration by parts to evaluate the integrals. $$ \int_{1}^{4} \ln \sqrt{x} d x $$

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

Use substitution to evaluate the indefinite integrals. $$ \int \frac{x-1}{1+4 x-2 x^{2}} d x $$

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 \(x\). $$ f(x)=\cos x, a=\frac{\pi}{6}, n=3 ; x=\frac{\pi}{7} $$

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

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

Use integration by parts to evaluate the integrals. $$ \int_{1}^{4} \sqrt{x} \ln \sqrt{x} d x $$

Use substitution to evaluate the indefinite integrals. $$ \int \frac{x^{2}-1}{x^{3}-3 x+1} d x $$

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 \(x\). $$ f(x)=x^{1 / 5}, a=-1, n=3 ; x=-0.9 $$

In Problems , use partial-fraction decompositon to evaluate each integral. $$ \int \frac{3 x^{2}+4 x+3}{\left(x^{2}+1\right)^{2}} d x $$

Use integration by parts to evaluate the integrals. $$ \int_{0}^{1} x e^{-x} d x $$

Use substitution to evaluate the indefinite integrals. $$ \int \frac{2 x}{1+2 x^{2}} d x $$

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 \(x\). $$ f(x)=e^{x}, a=2, n=3 ; x=2.1 $$

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

In Problems 23-26, complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}-2 x+2} d x $$

Use integration by parts to evaluate the integrals. $$ \int_{0}^{3} x^{2} e^{-x} d x $$

Use substitution to evaluate the indefinite integrals. $$ \int \frac{x^{3}-1}{x^{4}-4 x} d x $$

how that $$ T^{4} \approx T_{a}^{4}+4 T_{a}^{3}\left(T-T_{a}\right) $$ for \(T\) close to \(T_{a}\)

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

In Problems , complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}+4 x+5} d x $$

Use integration by parts to evaluate the integrals. $$ \int_{0}^{\pi / 3} e^{x} \sin x d x $$

Use substitution to evaluate the indefinite integrals. $$ \int 3 x e^{x^{2}} d x $$

Show that, for positive constants \(r\) and \(k\), $$ r N\left(1-\frac{N}{K}\right) \approx r N $$ for \(N\) close to 0 .

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

In Problems , complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}-4 x+13} d x $$

Use integration by parts to evaluate the integrals. $$ \int_{0}^{\pi / 6} e^{x} \cos x d x $$

Use substitution to evaluate the indefinite integrals. $$ \int \cos x e^{\sin x} d x $$

(a) Show that, for positive constants \(a\) and \(k\), $$ f(R)=\frac{a R}{k+R} \approx \frac{a}{k} R $$ for \(R\) close to 0 (b) Show that, for positive constants \(a\) and \(k\), $$ f(R)=\frac{a R}{k+R} \approx \frac{a}{2}+\frac{a}{4 k}(R-k) $$ for \(R\) close to \(k\).

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

In Problems , complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}+2 x+5} d x $$

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

Use substitution to evaluate the indefinite integrals. $$ \int \frac{1}{x} \csc ^{2}(\ln x) d x $$

In Problems 27-30, use the following form of the error term $$ R_{n+1}(x)=\frac{f^{(n+1)}(c)}{(n+1) !} x^{n+1} $$ where \(c\) is between 0 and \(x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([0, x] .\) (Do not compute the Taylor polynomial.) $$ f(x)=e^{x}, x=2, \text { error }<10^{-3} $$

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