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Math Practice

U-Substitution Practice Problems

Choose u as the inside function whose derivative appears as a factor, rewrite in terms of u and du, integrate, and substitute back. Add + C to indefinite integrals. Type functions with parentheses, as in sin(5x), e^(3x), ln(x) or sqrt(x^2 + 1).

  1. 1.

    Find .

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    Let , so . Then .

  2. 2.

    Find .

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    Let , so . Then .

  3. 3.

    Find .

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    Let , so and . Then .

  4. 4.

    Find .

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    Let , so . Then .

  5. 5.

    Find .

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    Let , so and . Then .

  6. 6.

    Find .

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    Let , so . Then . The absolute value isn't needed, because .

  7. 7.

    Use the substitution to find .

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    With , , so . (Using gives , which differs only by a constant, so it's also correct.)

  8. 8.

    Find .

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    Let , so . Then .

  9. 9.

    Find .

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    Let , so . Then .

  10. 10.

    Evaluate .

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    Let , so . The limits become and . Then .

  11. 11.

    Evaluate .

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    Let , . The limits become and . Then .

  12. 12.

    Evaluate .

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    Let , . The limits become and . Then .
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U-Substitution Practice Problems

These problems follow the u-substitution lesson. Problems 1–9 are indefinite integrals; 10–12 are definite integrals where you change the limits.

The method

  1. Choose , an inside function whose derivative appears as a factor.
  2. Find and solve for the piece of the integrand it replaces.
  3. Rewrite the integral entirely in , integrate, and substitute back (or change the limits).
  4. Add to indefinite integrals, and check by differentiating.

Common mistakes to watch for

  • Dropping the constant factor from , such as the in .
  • Leaving an in the integral after substituting.
  • Using the old limits after changing to .

Frequently Asked Questions

Why was my answer marked wrong when I forgot + C?+

An indefinite integral is a family of functions, so the + C is part of the answer. The three definite integrals at the end are numbers and need no C.

My answer looks different from the given one. Is it wrong?+

Not necessarily. Equivalent forms are accepted, such as (1/5)(x^2+3)^5 + C. Two antiderivatives can also differ by a constant, like sin(x)^2/2 and -cos(x)^2/2 in problem 7; both are accepted there.

More practice problems

Antiderivatives Practice Problems Fundamental Theorem of Calculus Practice Problems Chain Rule Practice Problems