Chapter 8: Techniques of IntegrationContent check — Calculus (Guichard), Chapter 8: Techniques of Integration

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int {(1-t)}^{9}dt}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int {(x^2+1)}^{2}dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int x{(x^2+1)}^{100}dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int \frac{1}{\sqrt[3]{1-5t}}dt}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int {\sin }^{3}x\cos xdx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int x\sqrt{100-x^2}dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int \frac{x^2}{\sqrt{1-x^3}}dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int \cos (\pi t)\cos (\sin (\pi t))dt}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int \frac{\sin x}{{\cos }^{3}x}dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int \tan xdx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle {\int }_0^{\pi }{\sin }^5(3x)\cos (3x)dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int {\sec }^{2}x\tan xdx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle {\int }_0^{\sqrt{\pi }/2}x{\sec }^2(x^2)\tan (x^2)dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int \frac{\sin (\tan x)}{{\cos }^{2}x}dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle {\int }_3^4\frac{1}{{(3x-7)}^2}dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle {\int }_0^{\pi /6}({\cos }^{2}x-{\sin }^{2}x)dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int \frac{6x}{{(x^2-7)}^{1/9}}dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle {\int }_{-1}^1(2x^3-1){(x^4-2x)}^{6}dx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle {\int }_{-1}^1{\sin }^{7}xdx}$

Find the antiderivatives or evaluate the definite integral in each problem.

${\displaystyle \int f(x)f^{\prime }(x)dx}$

Find the antiderivatives.

${\displaystyle \int {\sin }^{2}xdx}$

Find the antiderivatives.

${\displaystyle \int {\sin }^{3}xdx}$

Find the antiderivatives.

${\displaystyle \int {\sin }^{4}xdx}$

Find the antiderivatives.

${\displaystyle \int {\cos }^{2}x{\sin }^{3}xdx}$

Find the antiderivatives.

${\displaystyle \int {\cos }^{3}xdx}$

Find the antiderivatives.

${\displaystyle \int {\sin }^{2}x{\cos }^{2}xdx}$

Find the antiderivatives.

${\displaystyle \int {\cos }^{3}x{\sin }^{2}xdx}$

Find the antiderivatives.

${\displaystyle \int \sin x{(\cos x)}^{3/2}dx}$

Find the antiderivatives.

${\displaystyle \int {\sec }^{2}x{\csc }^{2}xdx}$

Find the antiderivatives.

${\displaystyle \int {\tan }^{3}x\sec xdx}$

Find the antiderivatives.

${\displaystyle \int \csc xdx}$

Find the antiderivatives.

${\displaystyle \int {\csc }^{3}xdx}$

Find the antiderivatives.

${\displaystyle \int \sqrt{x^2-1}dx}$

Find the antiderivatives.

${\displaystyle \int \sqrt{9+4x^2}dx}$

Find the antiderivatives.

${\displaystyle \int x\sqrt{1-x^2}dx}$

Find the antiderivatives.

${\displaystyle \int x^2\sqrt{1-x^2}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{1}{\sqrt{1+x^2}}dx}$

Find the antiderivatives.

${\displaystyle \int \sqrt{x^2+2x}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{1}{x^2(1+x^2)}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{x^2}{\sqrt{4-x^2}}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{\sqrt{x}}{\sqrt{1-x}}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{x^3}{\sqrt{4x^2-1}}dx}$

Find the antiderivatives.

${\displaystyle \int x\cos xdx}$

Find the antiderivatives.

${\displaystyle \int x^2\cos xdx}$

Find the antiderivatives.

${\displaystyle \int x{e}^{x}dx}$

Find the antiderivatives.

${\displaystyle \int x{e}^{x^2}dx}$

Find the antiderivatives.

${\displaystyle \int {\sin }^{2}xdx}$

Find the antiderivatives.

${\displaystyle \int \ln xdx}$

Find the antiderivatives.

${\displaystyle \int x\arctan xdx}$

Find the antiderivatives.

${\displaystyle \int x^3\sin xdx}$

Find the antiderivatives.

${\displaystyle \int x^3\cos xdx}$

Find the antiderivatives.

${\displaystyle \int x{\sin }^{2}xdx}$

Find the antiderivatives.

${\displaystyle \int x\sin x\cos xdx}$

Find the antiderivatives.

${\displaystyle \int \arctan (\sqrt{x})dx}$

Find the antiderivatives.

${\displaystyle \int \sin (\sqrt{x})dx}$

Find the antiderivatives.

${\displaystyle \int {\sec }^{2}x{\csc }^{2}xdx}$

Find the antiderivatives.

${\displaystyle \int \frac{1}{4-x^2}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{x^4}{4-x^2}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{1}{x^2+10x+25}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{x^2}{4-x^2}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{x^4}{4+x^2}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{1}{x^2+10x+29}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{x^3}{4+x^2}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{1}{x^2+10x+21}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{1}{2x^2-x-3}dx}$

Find the antiderivatives.

${\displaystyle \int \frac{1}{x^2+3x}dx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_1^{3}xdx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_0^3{x}^{2}dx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_2^4{x}^{3}dx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_1^3\frac{1}{x}dx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_1^2\frac{1}{1+x^2}dx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_0^{1}x\sqrt{1+x}dx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_1^5\frac{x}{1+x}dx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_0^1\sqrt{x^3+1}dx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_0^1\sqrt{x^4+1}dx}$

In the following problems, compute the trapezoid and Simpson approximations using 4 subintervals, and compute the error estimate for each. (Finding the maximum values of the second and fourth derivatives can be challenging for some of these; you may use a graphing calculator or computer software to estimate the maximum values.

${\displaystyle {\int }_1^4\sqrt{1+1/x}dx}$

${\displaystyle \int {(t+4)}^{3}dt}$

${\displaystyle \int t{(t^2-9)}^{3/2}dt}$

${\displaystyle \int (e^{t^2}+16)t{e}^{t^2}dt}$

${\displaystyle \int \sin t\cos 2tdt}$

${\displaystyle \int \tan t{\sec }^{2}tdt}$

${\displaystyle \int \frac{2t+1}{t^2+t+3}dt}$

${\displaystyle \int \frac{1}{t(t^2-4)}dt}$

${\displaystyle \int \frac{1}{{(25-t^2)}^{3/2}}dt}$

${\displaystyle \int \frac{\cos 3t}{\sqrt{\sin 3t}}dt}$

${\displaystyle \int t{\sec }^{2}tdt}$

${\displaystyle \int \frac{e^t}{\sqrt{e^t+1}}dt}$

${\displaystyle \int {\cos }^{4}tdt}$

${\displaystyle \int \frac{1}{t^2+3t}dt}$

${\displaystyle \int \frac{1}{t^2\sqrt{1+t^2}}dt}$

${\displaystyle \int \frac{{\sec }^{2}t}{{(1+\tan t)}^3}dt}$

${\displaystyle \int t^3\sqrt{t^2+1}dt}$

${\displaystyle \int e^t\sin tdt}$

${\displaystyle \int {(t^{3/2}+47)}^3\sqrt{t}dt}$

${\displaystyle \int \frac{t^3}{{(2-t^2)}^{5/2}}dt}$

${\displaystyle \int \frac{1}{t(9+4t^2)}dt}$

${\displaystyle \int \frac{\arctan 2t}{1+4t^2}dt}$

${\displaystyle \int \frac{t}{t^2+2t-3}dt}$

${\displaystyle \int {\sin }^{3}t{\cos }^{4}tdt}$

${\displaystyle \int \frac{1}{t^2-6t+9}dt}$

${\displaystyle \int \frac{1}{t{(\ln t)}^2}dt}$

${\displaystyle \int t{(\ln t)}^{2}dt}$

${\displaystyle \int t^3{e}^{t}dt}$

${\displaystyle \int \frac{t+1}{t^2+t-1}dt}$