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Integration by substitution

This method can be thought of as the reverse of the chain rule. To find \( \int_{a}^{b} f(g(x)) \mathrm{d}x \), we first substitute \( u = g(x)
\). We differentiate this to obtain \( \mathrm{d}u = g'(x) \mathrm{d}x \). Next, we change the limits, so the integral becomes \( \int_{g(a)}^{g(b)} f(u)
\mathrm{d}u \). If the appropriate substitution was made, it should be possible to evaluate this integral directly. In the case of indefinite integration, we do the same and then substitute back for \( u \) in the result.

This method can be used to find, for example, \( \int_{1}^{2}
\frac{\mathrm{d}x}{2x + 3} \). We first substitute \( u = 2x + 3 \) so that \(
\mathrm{d}u = 2\mathrm{d}x, \mathrm{d}x = \frac{1}{2} \mathrm{d}u \). Changing the limits, and substituting into the original integral gives \( \frac{1}{2} \int_{5}^{7} \frac{\mathrm{d}u}{u}
= \frac{1}{2} ( \ln{7} - \ln{5} ) = \frac{1}{2} \ln {\frac{7}{5}} \).

Alexander Frolkin 2001-03-13