Integral Calculus Problem Sets
Practice problems and worked solutions copied with their original problem headings preserved.
Practice problems and worked solutions copied with their original problem headings preserved.
1. $$\int (8x^3 - 2x + \sqrt{5})\,dx$$
2. $$\int (3 - 2x)(4x + 5)\,dx$$
3. $$\int \frac{(x - 2)^3}{3x^2}\,dx$$
4. $$\int (x^{3/4} + 2x^{1/2})^2\,dx$$
5. $$\int \sqrt{x^{1/2} - 2x^{3/2} + x^{5/2}}\,dx$$
6. $$\int \left(\frac{4}{x^2} - \frac{3}{x^3} + \frac{2}{x^4}\right)\,dx$$
7. $$\int \sqrt[3]{t}(\sqrt{t} + 1)^2\,dt$$
8. $$\int \frac{(5 - 2\sqrt{t})^2}{\sqrt[4]{t^3}}\,dt$$
9. $$\int \frac{x^{4/3} - 16}{x^{1/3} + 2}\,dx$$
10. $$\int \left(4 - \frac{3}{\sqrt{x}}\right)^2 \frac{dx}{\sqrt{x}}$$
11. $$\int (8x^3 - 3x^2 + 5x + 3)\,dx$$
12. $$\int_{2}^{3} \left(1 - \frac{x}{2}\right)^2\,dx$$
13. $$\int_{2}^{8} \frac{dy}{y\sqrt{2y}}$$
14. $$\int_{1}^{9} \sqrt{x(4 - x)}\,dx$$
15. $$\int_{1}^{4} \left(4x - \frac{5}{x\sqrt{x}}\right)\,dx$$
1. $$\int (8x^3 - 2x + \sqrt{5})\,dx = 2x^4 - x^2 + \sqrt{5}\,x + C$$
2. $$\int (3 - 2x)(4x + 5)\,dx = -\frac{8}{3}x^3 + x^2 + 15x + C$$
3. $$\int \frac{(x - 2)^3}{3x^2}\,dx = \frac{x^2}{6} - 2x + 4\ln|x| + \frac{8}{3x} + C$$
4. $$\int (x^{3/4} + 2x^{1/2})^2\,dx = \frac{2}{5}x^{5/2} + \frac{16}{9}x^{9/4} + 2x^2 + C$$
5. $$\int \sqrt{x^{1/2} - 2x^{3/2} + x^{5/2}}\,dx = \int x^{1/4}|x-1|\,dx$$
$$= \begin{cases} \frac{4}{9}x^{9/4} - \frac{4}{5}x^{5/4} + C, & x\ge 1 \\ \frac{4}{5}x^{5/4} - \frac{4}{9}x^{9/4} + C, & 0\le x\le 1 \end{cases} $$
6. $$\int \left(\frac{4}{x^2} - \frac{3}{x^3} + \frac{2}{x^4}\right)\,dx = -\frac{4}{x} + \frac{3}{2x^2} - \frac{2}{3x^3} + C$$
7. $$\int \sqrt[3]{t}(\sqrt{t} + 1)^2\,dt = \frac{3}{7}t^{7/3} + \frac{12}{11}t^{11/6} + \frac{3}{4}t^{4/3} + C$$
8. $$\int \frac{(5 - 2\sqrt{t})^2}{\sqrt[4]{t^3}}\,dt = 100t^{1/4} - \frac{80}{3}t^{3/4} + \frac{16}{5}t^{5/4} + C$$
9. $$\int \frac{x^{4/3} - 16}{x^{1/3} + 2}\,dx = \frac{1}{2}x^2 - \frac{6}{5}x^{5/3} + 3x^{4/3} - 8x + C$$
10. $$\int \left(4 - \frac{3}{\sqrt{x}}\right)^2 \frac{dx}{\sqrt{x}} = 32\sqrt{x} - 24\ln x - \frac{18}{\sqrt{x}} + C$$
11. $$\int (8x^3 - 3x^2 + 5x + 3)\,dx = 2x^4 - x^3 + \frac{5}{2}x^2 + 3x + C$$
12. $$\int_{2}^{3} \left(1 - \frac{x}{2}\right)^2\,dx = \frac{1}{12}$$
13. $$\int_{2}^{8} \frac{dy}{y\sqrt{2y}} = \frac{1}{2}$$
14. $$\int_{1}^{9} \sqrt{x}(4 - x)\,dx = -\frac{412}{15}$$
15. $$\int_{1}^{4} \left(4x - \frac{5}{x\sqrt{x}}\right)\,dx = 25$$
1. $$\int 5x(4 - 2x^2)^4\,dx$$
2. $$\int \frac{5x^2\,dx}{(1 - 2x^3)^6}$$
3. $$\int \frac{5y - 10}{y^2 - 4y + 11}\,dy$$
4. $$\int \frac{x\,dx}{4x^4 - 4x^2 + 1}$$
5. $$\int_{0}^{1} \frac{\tan^{-1} z}{z^2 + 1}\,dz$$
6. $$\int \frac{dz}{(2z - 3)^{3/2}}$$
7. $$\int (1 + 4e^{3x})^3 e^{3x}\,dx$$
8. $$\int \frac{2x^2 - 7x - 5}{2x + 1}\,dx$$
9. $$\int \frac{w + 1}{5w + 11}\,dw$$
10. $$\int \frac{x}{3x - 4}\,dx$$
11. $$\int \frac{x^5 - x^3 + 2x}{x^2 + 4}\,dx$$
12. $$\int \frac{e^{2x} - 1}{e^{2x} + 1}\,dx$$
13. $$\int \frac{dx}{x\ln^2 x}$$
14. $$\int_{0}^{\pi/2} \frac{\cos\beta\,d\beta}{\sqrt{1 + \sin\beta}}$$
15. $$\int \frac{\sin 2x\,dx}{\sqrt{\cos 2x - 1}}$$
16. $$\int \frac{dx}{x^{2/3}(\sqrt[3]{x}+5)^{3/2}}$$
17. $$\int \frac{e^x \sec^2(e^x + 1)}{\tan(e^x + 1)}\,dx$$
18. $$\int \frac{2e^{5x}}{3 - e^{5x}}\,dx$$
1. $$\int 5x(4 - 2x^2)^4\,dx = -\frac{1}{4}(4 - 2x^2)^5 + C$$
2. $$\int \frac{5x^2}{(1 - 2x^3)^6}\,dx = \frac{1}{6(1 - 2x^3)^5} + C$$
3. $$\int \frac{5y - 10}{y^2 - 4y + 11}\,dy = \frac{5}{2}\ln(y^2 - 4y + 11) + C$$
4. $$\int \frac{x}{4x^4 - 4x^2 + 1}\,dx = \frac{1}{4}\ln(4x^4 - 4x^2 + 1) + C$$
5. $$\int_{0}^{1} \frac{\tan^{-1} z}{z^2 + 1}\,dz = \frac{\pi^2}{32}$$
6. $$\int \frac{dz}{(2z - 3)^{3/2}} = -\frac{1}{\sqrt{2z - 3}} + C$$
7. $$\int (1 + 4e^{3x})^3 e^{3x}\,dx = \frac{(1 + 4e^{3x})^4}{48} + C$$
8. $$\int \frac{2x^2 - 7x - 5}{2x + 1}\,dx = x^2 - 4x - \frac{1}{2}\ln|2x + 1| + C$$
9. $$\int \frac{w + 1}{5w + 11}\,dw = \frac{5w-6\ln(5w+11)}{25} + C$$
10. $$\int \frac{x}{3x - 4}\,dx = \frac{1}{3}x + \frac{4}{9}\ln|3x - 4| + C$$
11. $$\int \frac{x^5 - x^3 + 2x}{x^2 + 4}\,dx = \frac{1}{4}x^4 - \frac{5}{2}x^2 + 11\ln(x^2 + 4) + C$$
12. $$\int \frac{e^{2x} - 1}{e^{2x} + 1}\,dx = \ln(e^{2x} + 1) - x + C$$
13. $$\int \frac{dx}{x\ln^2 x} = -\frac{1}{\ln x} + C$$
14. $$\int_{0}^{\pi/2} \frac{\cos\beta}{\sqrt{1 + \sin\beta}}\,d\beta = 2(\sqrt{2} - 1)$$
15. $$\int \frac{\sin 2x}{\sqrt{\cos 2x - 1}}\,dx = -\sqrt{\cos 2x - 1} + C$$
16. $$\int \frac{dx}{x^{2/3}(\sqrt[3]{x}+5)^{3/2}} = -\frac{6}{\sqrt{\sqrt[3]{x}+5}} + C$$
17. $$\int \frac{e^x \sec^2(e^x + 1)}{\tan(e^x + 1)}\,dx = \ln|\tan(e^x + 1)| + C$$
18. $$\int \frac{2e^{5x}}{3 - e^{5x}}\,dx = -\frac{2}{5}\ln|e^{5x}-3| + C$$
1. $$\int 3e^{5x}\,dx$$
2. $$\int 3^{\tan(4x)}\sec^{2}(4x)\,dx$$
3. $$\int \frac{dy}{\sqrt[3]{e^{2y}}}$$
4. $$\int e^{4-3x}\,dx$$
5. $$\int \frac{e^{3x}}{e^{x}-1}\,dx$$
6. $$\int x^{-2}e^{1/x}\,dx$$
7. $$\int \frac{5^{x}}{e^{2x}}\,dx$$
8. $$\int e^{3x}\sqrt[3]{\,4-e^{3x}\,}\,dx$$
9. $$\int \frac{3-4e^{5x}}{e^{2x}}\,dx$$
10. $$\int \left(e^{2x}-e^{-3x}\right)^{2}\,dx$$
11. $$\int \frac{5^{\cot(x/3)}}{\sin^{2}(x/3)}\,dx$$
12. $$\int 8^{\,1-2x^{2}}\,x\,dx$$
13. $$\int_{1}^{3}\sqrt[3]{5^{2y}}\,dy$$
14. $$\int_{0}^{3}\left(4-e^{-z}\right)^{2}\,dz$$
15. $$\int_{0}^{2}\left(e^{2x}+xe\right)\,dx$$
16. $$\int \frac{5^{2x}}{(5^{2x}-1)^{3}}\,dx$$
1. $$\int 3e^{5x}\,dx=\frac{3}{5}e^{5x}+C$$
2. $$\int 3^{\tan(4x)}\sec^{2}(4x)\,dx=\frac{3^{\tan(4x)}}{4\ln 3}+C$$
3. $$\int \frac{dy}{\sqrt[3]{e^{2y}}}=-\frac{3}{2}e^{-2y/3}+C$$
4. $$\int e^{4-3x}\,dx=-\frac{1}{3}e^{4-3x}+C$$
5. $$\int \frac{e^{3x}}{e^{x}-1}\,dx=\frac{1}{2}e^{2x}+e^{x}+\ln|e^{x}-1|+C$$
6. $$\int x^{-2}e^{1/x}\,dx=-e^{1/x}+C$$
7. $$\int \frac{5^{x}}{e^{2x}}\,dx=\frac{5^{x}}{(\ln 5-2)e^{2x}}+C$$
8. $$\int e^{3x}\sqrt[3]{\,4-e^{3x}\,}\,dx=-\frac{1}{4}\left(4-e^{3x}\right)^{4/3}+C$$
9. $$\int \frac{3-4e^{5x}}{e^{2x}}\,dx=-\frac{3}{2}e^{-2x}-\frac{4}{3}e^{3x}+C$$
10. $$\int \left(e^{2x}-e^{-3x}\right)^{2}\,dx=\frac{1}{4}e^{4x}+2e^{-x}-\frac{1}{6}e^{-6x}+C$$
11. $$\int \frac{5^{\cot(x/3)}}{\sin^{2}(x/3)}\,dx=-\frac{3}{\ln 5}\,5^{\cot(x/3)}+C$$
12. $$\int 8^{\,1-2x^{2}}\,x\,dx=-\frac{8^{\,1-2x^{2}}}{4\ln 8}+C$$
13. $$\int_{1}^{3}\sqrt[3]{5^{2y}}\,dy=\frac{3}{2\ln 5}\left(25-5^{2/3}\right)$$
14. $$\int_{0}^{3}\left(4-e^{-z}\right)^{2}\,dz=\frac{81}{2}+8e^{-3}-\frac{1}{2}e^{-6}$$
15. $$\int_{0}^{2}\left(e^{2x}+xe\right)\,dx=\frac{e^{4}-1}{2}+2e$$
16. $$\int \frac{5^{2x}}{(5^{2x}-1)^{3}}\,dx=-\frac{1}{4\ln 5}\cdot\frac{1}{(5^{2x}-1)^{2}}+C$$
Determine the area bounded by the curve $y^2=\frac{9x}{5}$ and the line $y=x-2$. Use horizontal strips.
From $y=x-2$, $x_1=y+2$. From $y^2=\frac{9x}{5}$, $x_2=\frac{5}{9}y^2$. The intersections are $(0.8,-1.2)$ and $(5,3)$.
$$A=\int_{-1.2}^{3}\left[(y+2)-\frac{5}{9}y^2\right]dy=\boxed{6.86\text{ sq. units}}.$$
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Find the area under $y=4-x^2$ from $x=-1$ to $x=2$.
Use direct integration:
$$A=\int_{-1}^{2}(4-x^2)\,dx=\left[4x-\frac{x^3}{3}\right]_{-1}^{2}=\boxed{9\text{ sq. units}}.$$
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A solid has a circular base of diameter $40$ cm. Every section perpendicular to a fixed diameter is an isosceles right triangle. Find the volume.
With radius $20$, let $y^2=400-x^2$. Following the board setup, compute one side and double by symmetry:
$$V=2\int_0^{20}y^2\,dx=2\int_0^{20}(400-x^2)\,dx=\boxed{10666.67\text{ cu. cm}}.$$
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A hemispherical tank of radius $20$ ft is filled with water to a depth of $15$ ft. Determine the work done in pumping all the water to the top of the tank. Use $62.4$ lb/ft$^3$ for water.
Using horizontal slices measured from the center, the water occupies $5\le y\le20$ and $x^2=400-y^2$.
$$W=62.4\pi\int_5^{20}y(400-y^2)\,dy=\boxed{6,981,869\text{ ft-lb}}.$$
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A parabola has equation $x^2=4y$. Determine the volume of the area bounded by the curve, the line $x=4$, and the $x$-axis when revolved about the $x$-axis.
The bounded parabolic segment has base $4$ and height $4$, so $A=\frac{ab}{3}=\frac{16}{3}$. Its centroid is $\bar{y}=\frac{3}{10}(4)=1.2$ from the $x$-axis.
By Pappus' theorem,
$$V=2\pi \bar{y}A=2\pi(1.2)\left(\frac{16}{3}\right)=\boxed{\frac{64\pi}{5}}.$$
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