For many integrals, no single antiderivative rule applies directly, and one must transform the integral into a more tractable form. The most common such technique is u-substitution, which reverses the Chain Rule. One identifies a portion of the integrand as \(u = g(x)\), computes the differential \(du = g'(x) \, dx\), and rewrites the integral as \(\int f(u) \, du\). After integrating in \(u\), the result is expressed in terms of \(x\). For example, \(\int 2x \cos(x^2) \, dx\) becomes \(\int \cos(u) \, du\) with \(u = x^2\), giving \(\sin(x^2) + C\). For definite integrals, the limits of integration must also be converted when performing substitution; if \(u = g(x)\), then the original limits \(x = a\) and \(x = b\) become \(u = g(a)\) and \(u = g(b)\), eliminating the need to substitute back at the end.
When the integrand is a product and direct integration is difficult, integration by parts is the technique of choice. Derived from the product rule for differentiation, it states that \(\int u \, dv = uv - \int v \, du\). The goal is to choose \(u\) and \(dv\) so that the new integral \(\int v \, du\) is simpler than the original. A useful guideline for selecting \(u\) is the LIATE rule, prioritizing in order: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions. The function higher on the list is typically chosen as \(u\). For instance, \(\int x e^x \, dx\) is best handled by setting \(u = x\) and \(dv = e^x \, dx\), giving \(\int x e^x \, dx = xe^x - e^x + C = e^x(x - 1) + C\). Similarly, \(\int \ln x \, dx = x \ln x - x + C\).
Sometimes two applications of parts are needed: integrating \(x^2 e^x\) requires parts twice, while integrals like \(\int e^x \sin x \, dx\) produce the original after two applications, allowing one to solve algebraically for the answer. For integrating rational functions \(P(x)/Q(x)\), the method of partial fractions is used. The denominator is factored into linear or irreducible quadratic pieces, and the original fraction is decomposed into a sum of simpler fractions with these as denominators; each piece is then integrated individually. Many trigonometric integrals can be reduced by identities, especially half-angle formulas. For example, \(\int \sin^2 x \, dx\) begins with the identity \(\sin^2 x = \frac{1 - \cos(2x)}{2}\), giving \(\frac{x}{2} - \frac{\sin(2x)}{4} + C\). Combining u-substitution, integration by parts, partial fractions, and trigonometric identities allows a wide variety of integrals to be evaluated.