Integration

Indefinite integral

1. Find each of following integrals:
   

   (a)   ${\displaystyle \int (x^3+2x+7)dx}$

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

   (c)   ${\displaystyle \int \frac{x^3+x^2-1}{x}dx}$

   Solutions:    (a)  $\cdots =\frac{1}{4}{x}^4+x^2+7x+C$;     (b)  $\cdots =x-\frac{1}{x}+\ln |x|+C$;     (c)  $\cdots =\frac{x^3}{3}+\frac{x^2}{2}-\ln |x|+C$
2. Integrate:
   

   (a)   ${\displaystyle \int (x+\sqrt{x}+1)dx}$

   (b)   ${\displaystyle \int \frac{3}{\sqrt{x}}dx}$

   (c)   ${\displaystyle \int (\sqrt[3]{x}+\sqrt[3]{x^2})dx}$

   (d)   ${\displaystyle \int \frac{2x^2-x^{-1/2}-1}{\sqrt{x}}dx}$

   Solutions:    (a)  $\cdots =\frac{1}{2}{x}^2+\frac{2}{3}\sqrt{x^3}+x+C$;     (b)  $\cdots =6\sqrt{x}+C$;     (c)  $\cdots =\frac{3}{4}\sqrt[3]{x^4}+\frac{3}{5}\sqrt[3]{x^5}+C$;     (d)  $\cdots =\frac{4}{5}\sqrt{x^5}-2\sqrt{x}-\ln |x|+C$
3. Integrate with respect to $x$:
   

   (a)   ${\displaystyle \int (\cos x-\sin x)dx}$

   (b)   ${\displaystyle \int (\sin x+3\cos x+e^x)dx}$

   Solutions:    (a)  $\cdots =\sin x+\cos x+C$;     (b)  $\cdots =-\cos x+3\sin x+e^x+C$
4. Find the equation of the function which has   $f^{\prime }(x)=2\cos x$   and   $f(0)=3$.
   Solution:    $f(x)=2\sin x+3$
5. Find the function $f$ knowing that its derivative is $f^{\prime }(x)=2x+3$ and its graph passes through the point $P(2,8)$.
   Solution:    $f(x)=x^2+3x-2$
6. A curve passes through the point $P(1,4)$ and its gradient function is $9x^2-5$. Write down the equation of this curve.
   Solution:    $y=3x^3-5x+6$
7. Find $y$ and express it in terms of $x$ (for $x>0$), given that ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{x}}$ and $y=3$ when $x=1$.
   Solution:    $y=\ln x+3$

Definite integral

8. Evaluate each of following integrals:
   

   (a)   ${\displaystyle \underset{-1}{\overset{2}{\int }}(-x^2+5)dx}$

   (b)   ${\displaystyle \underset{0}{\overset{9}{\int }}\sqrt{x}dx}$

   (c)   ${\displaystyle \underset{0}{\overset{\pi }{\int }}(\cos x+2)dx}$

   Solutions:    (a)  $\cdots =12$;     (b)  $\cdots =18$;     (c)  $\cdots =2\pi$
9. Evaluate the integrals:
   

   (a)   ${\displaystyle \underset{-2}{\overset{0}{\int }}(x^3-4x)dx}$

   (b)   ${\displaystyle \underset{0}{\overset{2}{\int }}(x^3-4x)dx}$

   (c)   ${\displaystyle \underset{-2}{\overset{2}{\int }}(x^3-4x)dx}$

   Solutions:    (a)  $\cdots =4$;     (b)  $\cdots =-4$;     (c)  $\cdots =0$
10. Find the area of the figure enclosed by the graph of the function $f(x)=x^2-x+1$, the $x$-axis and vertical lines $x=-1$ and $x=2$.
    Solution:    $A=\frac{9}{2}=4.5$
11. Find the area of the region between by the graph of the function $f(x)=e^x$ and the $x$-axis for $0⩽x⩽2$. Round the result to three significant figures.
    Solution:    $A=6.39$
12. Find the area of the region between by the curve $y=3\sqrt{x}$ and the $x$-axis on the interval $[1,4]$.
    Solution:    $A=14$
13. Find the area of the region between by the curve $y=9-x^2$ and the $x$-axis.
    Solution:    $A=36$
14. Find the area of the region enclosed by the graph of the function $f(x)=x^3+2x^2$ and the $x$-axis.
    Solution:    $A=\frac{4}{3}$
15. Find the area of the region enclosed by the curve $y=x^2$ and the straight line $y=x+2$.
    Solution:    $A=\frac{9}{2}=4.5$
16. Find the area between the curves $y=x^2-4x-2$ and $y=4-x^2$.
    Solution:    $A=21\frac{1}{3}$
17. Find the area of the figure between the graphs of the functions $f(x)=x^2-1$ and $g(x)=5-x^2$. Give the result in the exact form.
    Solution:    $A=8\sqrt{3}$
18. Find the area of the region enclosed by the graphs of the functions $f(x)=\sqrt{3x}$ and $g(x)=x^2-2x$.
    Solution:    $A=6$     (Hint: Use your GDC to determine the limits of integration.)

Integration by substitution

19. Integrate:
    

    (a)   ${\displaystyle \int (2x+3{)}^{2}dx}$

    (b)   ${\displaystyle \int \sqrt{2x+5}dx}$

    (c)   ${\displaystyle \int \frac{1}{x+5}dx}$

    Solutions:    (a)  $\cdots =\frac{1}{6}(2x+3{)}^3+C$;     (b)  $\cdots =\frac{1}{3}\sqrt{(2x+5{)}^3}+C$;     (c)  $\cdots =\ln |x+5|+C$
20. Integrate:
    

    (a)   ${\displaystyle \int e^{4x-1}dx}$

    (b)   ${\displaystyle \int \sin 5xdx}$

    (c)   ${\displaystyle \int \cos \frac{x+\pi }{7}dx}$

    Solutions:    (a)  $\cdots =\frac{1}{4}{e}^{4x-1}+C$;     (b)  $\cdots =-\frac{1}{5}\cos 5x+C$;     (c)  $\cdots =7\sin \frac{x+\pi }{7}+C$
21. Integrate:
    

    (a)   ${\displaystyle \int \frac{2xdx}{\sqrt{x^2+4}}}$

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

    (c)   ${\displaystyle \int \frac{\cos xdx}{\sin x}}$

    (d)   ${\displaystyle \int \tan xdx}$

    Solutions:    (a)  $\cdots =2\sqrt{x^2+4}+C$;     (b)  $\cdots =\ln (x^2+1)+C$;     (c)  $\cdots =\ln |\sin x|+C$;     (d)  $\cdots =-\ln |\cos x|+C$
22. Integrate:
    

    (a)   ${\displaystyle \int \frac{\sin x}{2\cos x+3}dx}$

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

    (c)   ${\displaystyle \int {\sin }^{5}x\cos xdx}$

    Solutions:    (a)  $\cdots =-\frac{1}{2}\ln (2\cos x+3)+C$;     (b)  $\cdots =-\frac{1}{\sin x}+C$;     (c)  $\cdots =\frac{1}{6}{\sin }^{6}x+C$
23. Integrate:
    

    (a)   ${\displaystyle \int \frac{x}{\sqrt{9-x^2}}dx}$

    (b)   ${\displaystyle \int \frac{2x-3}{x^2-3x+5}dx}$

    (c)   ${\displaystyle \int \frac{x+1}{\sqrt{x^2+2x+3}}dx}$

    Solutions:    (a)  $\cdots =-\sqrt{9-x^2}+C$;     (b)  $\cdots =\ln (x^2+3x+5)+C$;     (c)  $\cdots =\sqrt{x^2+2x+3}+C$
24. Evaluate the following integrals:
    

    (a)   ${\displaystyle \underset{1}{\overset{8}{\int }}\sqrt{3x+1}dx}$

    (b)   ${\displaystyle \underset{-1}{\overset{3}{\int }}\sqrt{2x+3}dx}$

    Solutions:    (a)  $\cdots =26$;     (b)  $\cdots =\frac{26}{3}=8\frac{2}{3}$
25. Evaluate the following integrals:
    

    (a)   ${\displaystyle \underset{1}{\overset{2}{\int }}(2x-1{)}^{3}dx}$

    (b)   ${\displaystyle \underset{-1}{\overset{12}{\int }}\sqrt[3]{2x+3}dx}$

    (c)   ${\displaystyle \underset{0}{\overset{5}{\int }}\frac{1}{\sqrt{2x+3}}dx}$

    Solutions:    (a)  $\cdots =10$;     (b)  $\cdots =30$;     (c)  $\cdots =2$
26. Evaluate the following integrals:
    

    (a)   ${\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin 2xdx}$

    (b)   ${\displaystyle \underset{0}{\overset{\pi }{\int }}\cos \frac{x-\pi }{4}dx}$

    (c)   ${\displaystyle \underset{-2}{\overset{0}{\int }}{e}^{x+2}dx}$

    Solutions:    (a)  $\cdots =1$;     (b)  $\cdots =2\sqrt{2}\approx 2.83$;     (c)  $\cdots =e^2-1\approx 6.39$
27. Find the area of the region between by the graph of the function $f(x)=\sqrt{2x+1}$ and the $x$-axis for $0⩽x⩽4$.
    Solution:    $A=8\frac{2}{3}$
28. Find the area of the region enclosed by the graph of the function $f(x)=\sqrt{x+4}$, the $x$-axis and the vertical line $x=5$.
    Solution:    $A=18$
29. Find the area of the region enclosed by the graph of the function $f(x)=\frac{12}{x+2}$ and the straight line $y=-x+6$. Round the result to three significant figures.
    Solution:    $A=16-12\ln 3\approx 2.82$

Improper integrals

30. Evaluate the integrals with infinite limits of integration:
    

    (a)   ${\displaystyle \underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{1}{x^2}dx}$

    (b)   ${\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{4}{(x+2{)}^2}dx}$

    (c)   ${\displaystyle \underset{-\mathrm{\infty }}{\overset{0}{\int }}{e}^{2x}dx}$

    Solutions:    (a)  $\cdots =1$;     (b)  $\cdots =2$;     (c)  $\cdots =\frac{1}{2}$
31. Find the area of the region between by the graph of the function $f(x)=x{e}^{-x^2}$ and the $x$-axis for $x\in [0,\mathrm{\infty })$.
    Solution:    $A=\frac{1}{2}$

Volume of the solid of revolution

32. The function has the equation $f(x)=\sqrt{x}$. Find the volume of the solid of revolution when the part of the graph between $x=0$ and $x=4$ is rotated through ${360}^{\circ }$ around the $x$-axis.
    Solution:    $V=8\pi$
33. The part of the function $f(x)=3-x$ between $x=0$ and $x=3$ is rotated by ${360}^{\circ }$ around the $x$-axis. Name the solid obtained this way and find its volume. Write the result in exact form.
    Solution:    It's a cone and its volume is $9\pi$.
34. Find the volume of revolution when the part of the curve $y=1-x^2$ between both $x$-intercepts is rotated by $2\pi$ radians around $x$-axis. Round the result to three significant figures.
    Solution:    $V\approx 3.35$
35. Find the volume of the solid formed when the graph of the function $f(x)=\sqrt{9-x^2}$ is revolved about the $x$-axis. Write the result as a multiple of $\pi$.
    Solution:    $V=36\pi$     (Hint: You need the limits of integration. To determine them you must find the interval where the function is defined.)

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