Evaluate the integral of 2x exp(x²) from 0 to sqrt(ln 5).
Answer and explanation
4
Set u=x²; the limits become 0 and ln 5. The integral is exp(ln 5)−1=4.
Subjects
University calculus
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Sample exercises
Evaluate the improper integral of 3/x² from 1 to infinity.
Answer and explanation
3
The antiderivative is −3/x. Its limit at infinity is 0, so the value is 0−(−3)=3.
Which integral from 1 to infinity converges?
- The integral of 1
- The integral of 1/sqrt(x)
- The integral of x^(−3/2)
- The integral of 1/x
Answer and explanation
The integral of x^(−3/2)
The exponent p=3/2 exceeds 1; the others have p≤1.
Evaluate the integral of x ln x from 1 to e. Enter the coefficient of e² in the result A e² + 1/4.
Answer and explanation
0.25
Integration by parts gives x² ln x/2−x²/4. Evaluation gives e²/4+1/4.
Explain why the integral of 1/x from −1 to 1 is not an ordinary convergent improper integral, even though the symmetric principal value is zero.
Answer and explanation
Each one-sided integral at zero diverges. Symmetric cancellation is a different limiting procedure, namely a principal value.
Award credit for separate endpoint limits, divergence, and distinguishing a principal value from ordinary improper convergence.
Evaluate the integral of 1/sqrt(x) from 0 to 9.
Answer and explanation
6
Use the lower limit ε>0: 2sqrt(9)−2sqrt(ε) tends to 6.
Study notes
Integral methods and improper limits
Substitution reverses a chain rule; integration by parts reverses a product rule.
UniversityIntegral methods and improper limits · revision notes
Substitution reverses a chain rule; integration by parts reverses a product rule.
UniversityDifferentiate a polynomial-exponential product
Differentiate before substituting. The value at zero can hide a nonzero product-rule term.
UniversityDifferentiate a polynomial-exponential product · formula reference
Differentiate before substituting. The value at zero can hide a nonzero product-rule term.
UniversityIntegrate a decaying exponential
Convergence requires k > 0. The upper limit is a limit, not a substitution of a finite number.
UniversityIntegrate a decaying exponential · formula reference
Convergence requires k > 0. The upper limit is a limit, not a substitution of a finite number.
UniversityIntegration by parts with a logarithm
Here n > −1. Treat x = 0 as an improper endpoint and use its limiting boundary term.
UniversityIntegration by parts with a logarithm · formula reference
Here n > −1. Treat x = 0 as an improper endpoint and use its limiting boundary term.
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