Chapter 8 Problems

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

Read the chapter problems → · Chapter solutions →

1. Original problem

   Calculate the definite integral ${\int }_0^{\mathrm{\infty }}{e}^{-3x}dx$

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2. Original problem

   Calculate the indefinite integral $\int \frac{x+3}{x^3}dx$

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3. Original problem

   Evaluate the indefinite integral $\int \frac{x^2+3x+3}{3x^4}dx$

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4. Original problem

   Calculate the definite integral ${\int }_0^{\mathrm{\infty }}3e^{\frac{-x}{2}}dx$

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5. Original problem

   Evaluate the definite integral ${\int }_{-0.5}^0-\frac{1}{\sqrt{1-4x^2}}dx$

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6. Original problem

   Evaluate the indefinite integral $\int \frac{1}{\sqrt{9x^2-12x+5}}dx$

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7. Original problem

   Evaluate the definite integral ${\int }_0^6\frac{1}{(4x-10)\sqrt{x^2-5x-1}}dx$

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8. Original problem

   Evaluate the indefinite integral $\int \frac{2x}{x^4-8x^2+17}dx$

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9. Original problem

   Evaluate the indefinite integral $\int \frac{2x}{\sqrt{x^4+3x^2+2}}dx$

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10. Original problem

    Evaluate the indefinite integral $\int \frac{1}{(2x-2)\sqrt{x}}dx$

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11. Added problem

    Find $\int (3x^2+4x-5)dx$.

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12. Added problem

    Evaluate ${{\int }_1}^{\sqrt{10}}2xdx$.

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13. Added problem

    Evaluate $\int 3/xdx$ on a real interval that avoids zero.

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14. Added problem

    Find $\int (2\sin x+3e^x)dx$.

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15. Added problem

    Evaluate ${{\int }_{-1}}^1{x}^{3}dx$ and explain the zero result.

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16. Added problem

    Evaluate ${{\int }_2}^{1}2xdx$ and explain its sign.

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17. Added problem

    Find F given $F\prime \left(x\right)=3x^2$ and $F\left(2\right)=12$.

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18. Added problem

    Evaluate ${{\int }_2}^{\infty }1/x^{3}dx$.

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19. Added problem

    Determine whether ${{\int }_1}^{\infty }1/xdx$ converges.

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20. Added problem

    Evaluate ${{\int }_1}^{\infty }{e}^{-x/2}dx$.

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21. Added problem

    Evaluate ${{\int }_0}^{1}1/\sqrt{x}dx$ as an improper integral.

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22. Added problem

    Find $\int 1/(x^2+4)dx$.

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23. Added problem

    Find $\int 1/\sqrt{9-x^2}dx$ on its interior real domain.

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24. Added problem

    Evaluate $\int 1/(x^2-4x+5)dx$.

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25. Added problem

    Evaluate ${{\int }_0}^{1}2x/(1+x^4)dx$.

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26. Added problem

    Rewrite ${{\int }_0}^{1/2}x/\sqrt{1-x^2}dx$ using $x=\sin \theta$. Do not evaluate it.

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27. Added problem

    For $F\left(x\right)={{\int }_0}^x\sqrt{1+t^2}dt$, find F′(x).

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28. Added problem

    Find the derivative of $F\left(x\right)={{\int }_0}^{x^2}\cos tdt$.

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29. Added problem

    A student writes $\int 2xdx=x^2$. What is missing? If the primitive must have F(0)=3, what is it?

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30. Added problem

    Evaluate ${{\int }_0}^{1/2}1/\sqrt{1-x^2}dx$.

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