Improper Integrals — Practice Problems and Worked Solutions

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16 worked problems

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1. 1.

   Evaluate the convergent improper integral.

   $${\int }_1^{\infty }\frac{1}{x^2}dx$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{Replace the improper endpoint with a limit.}$$
   2. $${\int }_1^{\infty }\frac{1}{x^2}dx=1$$
2. 2.

   Evaluate the convergent improper integral.

   $${\int }_1^{\infty }\frac{1}{x^3}dx$$

   Answer: $\frac{1}{2}$

   Step-by-step solution

   1. $$\text{Replace the improper endpoint with a limit.}$$
   2. $${\int }_1^{\infty }\frac{1}{x^3}dx=\frac{1}{2}$$
3. 3.

   Evaluate the convergent improper integral.

   $${\int }_0^{\infty }{e}^{-x}dx$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{Replace the improper endpoint with a limit.}$$
   2. $${\int }_0^{\infty }{e}^{-x}dx=1$$
4. 4.

   Evaluate the convergent improper integral.

   $${\int }_0^{\infty }x{e}^{-x}dx$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{Replace the improper endpoint with a limit.}$$
   2. $${\int }_0^{\infty }x{e}^{-x}dx=1$$
5. 5.

   Evaluate the convergent improper integral.

   $${\int }_0^1\frac{1}{\sqrt{x}}dx$$

   Answer: $2$

   Step-by-step solution

   1. $$\text{Replace the improper endpoint with a limit.}$$
   2. $${\int }_0^1\frac{1}{\sqrt{x}}dx=2$$
6. 6.

   Evaluate the convergent improper integral.

   $${\int }_0^1\sqrt[3]{x}dx$$

   Answer: $\frac{3}{4}$

   Step-by-step solution

   1. $$\text{Replace the improper endpoint with a limit.}$$
   2. $${\int }_0^1\sqrt[3]{x}dx=\frac{3}{4}$$
7. 7.

   Evaluate the convergent improper integral.

   $${\int }_1^{\infty }\frac{1}{x^4}dx$$

   Answer: $\frac{1}{3}$

   Step-by-step solution

   1. $$\text{Replace the improper endpoint with a limit.}$$
   2. $${\int }_1^{\infty }\frac{1}{x^4}dx=\frac{1}{3}$$
8. 8.

   Evaluate the convergent improper integral.

   $${\int }_1^{\infty }\frac{1}{x^5}dx$$

   Answer: $\frac{1}{4}$

   Step-by-step solution

   1. $$\text{Replace the improper endpoint with a limit.}$$
   2. $${\int }_1^{\infty }\frac{1}{x^5}dx=\frac{1}{4}$$
9. 9.

   Evaluate the convergent improper integral.

   $${\int }_0^{\infty }{e}^{-2x}dx$$

   Answer: $\frac{1}{2}$

   Step-by-step solution

   1. $$\text{Replace the improper endpoint with a limit.}$$
   2. $${\int }_0^{\infty }{e}^{-2x}dx=\frac{1}{2}$$
10. 10.

    Evaluate the convergent improper integral.

    $${\int }_0^{\infty }{e}^{-3x}dx$$

    Answer: $\frac{1}{3}$

    Step-by-step solution

    1. $$\text{Replace the improper endpoint with a limit.}$$
    2. $${\int }_0^{\infty }{e}^{-3x}dx=\frac{1}{3}$$
11. 11.

    Evaluate the convergent improper integral.

    $${\int }_0^1\frac{1}{\sqrt[3]{x}}dx$$

    Answer: $\frac{3}{2}$

    Step-by-step solution

    1. $$\text{Replace the improper endpoint with a limit.}$$
    2. $${\int }_0^1\frac{1}{\sqrt[3]{x}}dx=\frac{3}{2}$$
12. 12.

    Evaluate the convergent improper integral.

    $${\int }_0^1\sqrt{x}dx$$

    Answer: $\frac{2}{3}$

    Step-by-step solution

    1. $$\text{Replace the improper endpoint with a limit.}$$
    2. $${\int }_0^1\sqrt{x}dx=\frac{2}{3}$$
13. 13.

    Evaluate the convergent improper integral.

    $${\int }_2^{\infty }\frac{1}{x^2}dx$$

    Answer: $\frac{1}{2}$

    Step-by-step solution

    1. $$\text{Replace the improper endpoint with a limit.}$$
    2. $${\int }_2^{\infty }\frac{1}{x^2}dx=\frac{1}{2}$$
14. 14.

    Evaluate the convergent improper integral.

    $${\int }_0^{\infty }{x}^2{e}^{-x}dx$$

    Answer: $2$

    Step-by-step solution

    1. $$\text{Replace the improper endpoint with a limit.}$$
    2. $${\int }_0^{\infty }{x}^2{e}^{-x}dx=2$$
15. 15.

    Evaluate the convergent improper integral.

    $${\int }_0^1\frac{1}{x^{\frac{2}{3}}}dx$$

    Answer: $3$

    Step-by-step solution

    1. $$\text{Replace the improper endpoint with a limit.}$$
    2. $${\int }_0^1\frac{1}{x^{\frac{2}{3}}}dx=3$$
16. 16.

    Evaluate the convergent improper integral.

    $${\int }_2^{\infty }\frac{1}{x^3}dx$$

    Answer: $\frac{1}{8}$

    Step-by-step solution

    1. $$\text{Replace the improper endpoint with a limit.}$$
    2. $${\int }_2^{\infty }\frac{1}{x^3}dx=\frac{1}{8}$$

More from Calculus II

• Integration by Parts
• Trigonometric Integrals
• Partial Fractions
• Sequence Limits
• Geometric Series
• p-Series
• Convergence Tests
• Power Series
• Taylor Series
• Parametric Derivatives
• Parametric Arc Length
• Polar Area