Integration by substitution

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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