Integration

Indefinite integral

1. Find each of following integrals:
   

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

   (b)   \({\displaystyle \int \left(1+\frac{1}{x^2}+\frac{1}{x}\right)\,dx}\)

   (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)   \({\displaystyle \int (\sqrt[\scriptstyle3]{x}+\sqrt[\scriptstyle3]{x^2})\,dx }\)

   (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   \(f'(x)=2\cos x\)   and   \(f(0)=3\).
   Solution:    \(f(x)=2\sin x+3\)
5. Find the function \(f\) knowing that its derivative is \(f'(x)=2x+3\) 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 \(x\gt0\)), 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)   \({\displaystyle \int\limits_{-1}^2 (-x^2+5)~ dx}\)

   (b)   \({\displaystyle \int\limits_0^9 \sqrt{x}~ dx}\)

   (c)   \({\displaystyle \int\limits_0^\pi (\cos x+2)~dx }\)

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

   (a)   \({\displaystyle \int\limits_{-2}^0 (x^3-4x)~ dx}\)

   (b)   \({\displaystyle \int\limits_0^2 (x^3-4x)~ dx}\)

   (c)   \({\displaystyle \int\limits_{-2}^2 (x^3-4x)~ dx}\)

   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 \(0\leqslant x\leqslant 2\). 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)   \({\displaystyle \int \sin 5x\,dx}\)

    (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)   \({\displaystyle \int \frac{2x\,dx}{\sqrt{x^2+4}}}\)

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

    (c)   \({\displaystyle \int \frac{\cos x\,dx}{\sin x}}\)

    (d)   \({\displaystyle \int \tan x\,dx }\)

    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)   \({\displaystyle \int \sin^5 x \cos x\,dx}\)

    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)   \({\displaystyle \int\limits_1^8 \sqrt{3x+1}\,dx}\)

    (b)   \({\displaystyle \int\limits_{-1}^3 \sqrt{2x+3}\,dx}\)

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

    (a)   \({\displaystyle \int\limits_1^2 (2x-1)^3\,dx}\)

    (b)   \({\displaystyle \int\limits_{-1}^{12} \sqrt[\scriptstyle3]{2x+3}\,dx}\)

    (c)   \({\displaystyle \int\limits_0^5 \frac{1}{\sqrt{2x+3}}\,dx}\)

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

    (a)   \({\displaystyle \int\limits_0^\frac{\pi}{2} \sin 2x\,dx}\)

    (b)   \({\displaystyle \int\limits_0^\pi \cos\frac{x-\pi}{4}\,dx}\)

    (c)   \({\displaystyle \int\limits_{-2}^0 e^{x+2}\,dx}\)

    Solutions:    (a)  \(\cdots=1\);     (b)  \(\cdots=2\sqrt{2}\approx2.83\);     (c)  \(\cdots=e^2-1\approx6.39\)
27. Find the area of the region between by the graph of the function \(f(x)=\sqrt{2x+1}\) and the \(x\)-axis for \(0\leqslant x\leqslant 4\).
    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 \(f(x)=\frac{\textstyle 12}{\textstyle x+2}\) and the straight line \(y=-x+6\). Round the result to three significant figures.
    Solution:    \(A=16-12\ln3\approx2.82\)

Improper integrals

30. Evaluate the integrals with infinite limits of integration:
    

    (a)   \({\displaystyle \int\limits_1^\infty \frac{1}{x^2}\,dx}\)

    (b)   \({\displaystyle \int\limits_0^\infty \frac{4}{(x+2)^2}\,dx}\)

    (c)   \({\displaystyle \int\limits_{-\infty}^0 e^{2x}\,dx}\)

    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 \(x\in[0,\infty)\).
    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:    \(V\approx3.35\)
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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