Chapter 9 Problems

Thirty chapter problems: 10 from the original book and 20 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem

   Solve the indefinite integral $\int \csc xdx$

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2. Original problem

   Evaluate ${\int }_0^{\pi }\frac{\cos x}{\cos (\sin x)}dx$

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3. Original problem

   Evaluate ${\int }_0^{2}6\sqrt{4-\sqrt{x}}dx$

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4. Original problem

   Solve the indefinite integral $\int \frac{2}{3}\sqrt{9-x^2}dx$

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5. Original problem

   Evaluate the indefinite integral $\int \frac{1-4x^2}{x}dx$

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6. Original problem · clarified

   Evaluate ${\int }_0^{\mathrm{\infty }}\frac{2-x}{8-4x+2x^2-x^3}dx$

   Clarification: Interpret the removable hole at x=2 by taking the finite one-sided limits, or equivalently by using the continuous extension.

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7. Original problem

   Evaluate ${\int }_1^3\frac{x\cdot \sqrt{2-x+\frac{3}{x}}}{\sqrt{x}}dx$

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8. Original problem

   Evaluate the indefinite integral $\int \tan x\cdot \frac{\sin (\ln (\cos x))}{\cos (\ln (\cos x))+3}dx$

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9. Original problem

   Evaluate the indefinite integral $\int \frac{\tan (\ln x)}{x\cdot \ln (\cos (\ln x))+3x}dx$

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10. Original problem

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

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11. Added problem

    Evaluate $\int 2x(x^2+1{)}^{3}dx$.

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12. Added problem

    Evaluate $\int 2x/(x^2+4)dx$.

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13. Added problem

    Evaluate $\int x^2\sin \left(x^3\right)dx$.

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14. Added problem

    Find $\int e^x/(e^x+2)dx$.

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15. Added problem

    Evaluate $\int (\ln x{)}^2/xdx$ for $x>0$.

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16. Added problem

    Evaluate ${{\int }_0}^{2}2x/(x^2+1)dx$ using changed bounds.

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17. Added problem

    Evaluate ${{\int }_0}^{1}x/\sqrt{1-x^2}dx$ with u=1−x², treating the endpoint as a limit.

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18. Added problem

    Evaluate $\int (x^2+1)/(x-1)dx$.

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19. Added problem

    Use trig substitution to evaluate ${{\int }_0}^{1}1/(4-x^2{)}^{3/2}dx$.

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20. Added problem

    Evaluate ${{\int }_0}^4\sqrt{16-x^2}dx$ and confirm it geometrically.

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21. Added problem

    Evaluate ${{\int }_1}^2\sqrt{2x-x^2}dx$.

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22. Added problem

    Use x=3tanθ to evaluate ${{\int }_0}^{3}1/(x^2+9)dx$.

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23. Added problem

    Derive $\int \sec xdx$ by choosing u=sec x+tan x.

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24. Added problem

    Evaluate $\int \tan xdx$.

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25. Added problem

    Evaluate $\int {\sin }^2\left(2x\right)dx$.

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26. Added problem

    A student uses u=x² in $\int x{e}^{x^2}dx$ and writes $\int e^{u}du$. Diagnose and correct the error.

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27. Added problem

    Why is the angle interval important when using x=sinθ to simplify $\sqrt{1-x^2}$?

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28. Added problem

    Evaluate $\int \cos x/(3+\sin x)dx$.

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29. Added problem

    Evaluate ${{\int }_{-1}}^{1}2x\cos \left(x^2\right)dx$ using a primitive, and explain the sign cancellation.

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30. Added problem

    For $\int \sqrt{4-x^2}dx$, compare u=4−x² with x=2sinθ. Which choice gives a useful first step, and why?

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