Chapter 7

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{1}{\sqrt{4-x}} d x $$

Find an antiderivative. $$ p(x)=x^{2}-6 x+17 $$

Find the integrals. $$ \int_{3}^{5} x \cos x d x $$

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{2}^{5}\left(x^{3}-\pi x^{2}\right) d x $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{d y}{y+5} $$

Find an antiderivative. $$ q(z)=\sqrt{z} $$

Use integration by parts twice to evaluate the integral. $$ \int t^{2} e^{5 t} d t $$

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{1} e^{-0.2 t} d t $$

Find the integrals in problems. Check your answers by differentiation. $$ \int 12 x^{2} \cos \left(x^{3}\right) d x $$

Find an antiderivative. $$ f(x)=5 x-\sqrt{x} $$

Use integration by parts twice to evaluate the integral. $$ \int(\ln t)^{2} d t $$

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{1} 2 e^{x} d x $$

Find the integrals in problems. Check your answers by differentiation. $$ \int(2 t-7)^{73} d t $$

Find an antiderivative. $$ p(z)=(\sqrt{z})^{3} $$

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{-1}^{1} \cos t d t $$

Find the integrals in problems. Check your answers by differentiation. $$ \int\left(x^{2}+3\right)^{2} d x $$

Find an antiderivative. $$ h(z)=\frac{1}{z} $$

Both exactly [e.g. \(\ln (3 \pi)]\) and numerically [e.g. \(\ln (3 \pi) \approx 2.243]\). $$ \int_{0}^{10} z e^{-z} d z $$

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{\pi / 4}(\sin t+\cos t) d t $$

Find the integrals in problems. Check your answers by differentiation. $$ \int y^{2}(1+y)^{2} d y $$

Find an antiderivative. $$ p(t)=t^{3}-\frac{t^{2}}{2}-t $$

Both exactly [e.g. \(\ln (3 \pi)]\) and numerically [e.g. \(\ln (3 \pi) \approx 2.243]\). $$ \int_{1}^{3} t \ln t d t $$

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{3} e^{0.05 t} d t $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \sin (3-t) d t $$

Find an antiderivative. $$ g(z)=\frac{1}{z^{3}} $$

Both exactly [e.g. \(\ln (3 \pi)]\) and numerically [e.g. \(\ln (3 \pi) \approx 2.243]\). $$ \int_{0}^{5} \ln (1+t) d t $$

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{1}\left(6 q^{2}+4\right) d q $$

Find an antiderivative. $$ q(y)=y^{4}+\frac{1}{y} $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \sin \theta(\cos \theta+5)^{7} d \theta $$

For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate. Do not evaluate the integrals. (a) \(\int \frac{x^{2}}{1+x^{3}} d x\) (b) \(\int x e^{x^{2}} d x\) (c) \(\int x^{2} \ln \left(x^{3}+1\right) d x\) (d) \(\int \frac{1}{\sqrt{3 x+1}} d x\) (e) \(\int x^{2} \ln x d x\) (f) \(\int \ln x d x\)

Use substitution to express each of the following integrals as a multiple of \(\int_{a}^{b}(1 / w) d w\) for some \(a\) and \(b\). Then evaluate the integrals. (a) \(\int_{0}^{1} \frac{x}{1+x^{2}} d x\) (b) \(\int_{0}^{\pi / 4} \frac{\sin x}{\cos x} d x\)

Find an antiderivative. $$ f(x)=x^{6}-\frac{1}{7 x^{6}} $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \sqrt{\cos 3 t} \sin 3 t d t $$

Find \(\int_{1}^{2} \ln x d x\) numerically. Find \(\int_{1}^{2} \ln x d x\) using antiderivatives. Check that your answers agree.

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in problem. $$ \int_{0}^{2} x\left(x^{2}+1\right)^{2} d x $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{t}{1+3 t^{2}} d t $$

Find an antiderivative. $$ g(x)=\frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}} $$

Find the exact area. Under \(y=t e^{-t}\) for \(0 \leq t \leq 2\).

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in problem. $$ \int_{0}^{3} \frac{2 x}{x^{2}+1} d x $$

Find an antiderivative. $$ g(t)=e^{-3 t} $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \sin ^{6} \theta \cos \theta d \theta $$

Find the exact area. Between \(y=\ln x\) and \(y=\ln \left(x^{2}\right)\) for \(1 \leq x \leq 2\).

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in problem. $$ \int_{0}^{1} 2 t e^{-t^{2}} d t $$

Find the integrals in problems. Check your answers by differentiation. $$ \int x^{2} e^{x^{3}+1} d x $$

Find an antiderivative. $$ h(t)=\cos t $$

Find the exact area. Between \(f(t)=\ln \left(t^{2}-1\right)\) and \(g(t)=\ln (t-1)\) for \(2

Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in problem. $$ \int_{0}^{3} \frac{1}{\sqrt{t+1}} d t $$

Find an antiderivative. $$ g(t)=5+\cos t $$

Find the integrals in problems. Check your answers by differentiation. $$ \int \sin ^{2} x \cos x d x $$

Estimate \(\int_{0}^{10} f(x) g^{\prime}(x) d x\) if \(f(x)=x^{2}\) and \(g\) has the values in the following table. $$ \begin{array}{c|c|c|c|c|c|c} \hline x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline g(x) & 2.3 & 3.1 & 4.1 & 5.5 & 5.9 & 6.1 \\ \hline \end{array} $$