u-Substitution Math Example 2

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Example 2

medium
Evaluate ∫01xex2 dx\displaystyle\int_0^1 xe^{x^2}\,dx.

Solution

  1. 1
    Let u=x2u = x^2, du=2x dxdu = 2x\,dx, so x dx=du2x\,dx = \frac{du}{2}.
  2. 2
    New limits: x=0⇒u=0x=0 \Rightarrow u=0; x=1⇒u=1x=1 \Rightarrow u=1.
  3. 3
    12∫01eu du=12[eu]01=e−12\frac{1}{2}\int_0^1 e^u\,du = \frac{1}{2}[e^u]_0^1 = \frac{e-1}{2}.

Answer

e−12\frac{e-1}{2}
For definite integrals, change the limits to uu-values to avoid back-substitution. The factor 12\frac{1}{2} arises because x dx=du2x\,dx = \frac{du}{2}.

About u-Substitution

An integration technique where you substitute u=g(x)u = g(x) and du=g′(x) dxdu = g'(x)\,dx to transform a complicated integral into a simpler one. It is the reverse of the chain rule for differentiation.

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More u-Substitution Examples

Example 1 easy

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Example 3 easy

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Example 4 hard

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