Integrals, by the form of the integrand
| The integrand contains | Method |
|---|---|
| a function and its derivative | \(u\)-substitution |
| a product of functions of different types | integration by parts, with \(u\) chosen in the order logarithmic, inverse trigonometric, algebraic, trigonometric, exponential |
| \(\sqrt{a^2-x^2}\), \(\sqrt{a^2+x^2}\), \(\sqrt{x^2-a^2}\) | trigonometric substitution \(x=a\sin\theta\), \(x=a\tan\theta\), \(x=a\sec\theta\) respectively |
| a proper rational function | partial fractions |
| an odd power of \(\sin\) or \(\cos\) | factor out one \(\sin\) or \(\cos\), rewrite the rest with \(\sin^2+\cos^2=1\), and substitute |
| even powers of \(\sin\) and \(\cos\) only | the half-angle identities |
| a rational function \(R(\sin x, \cos x)\) | the Weierstrass substitution \(t = \tan(x/2)\) |
Symmetry. If \(f\) is odd, then \(\int_{-a}^a f = 0\); if \(f\) is even, then \(\int_{-a}^a f = 2\int_0^a f\).
Standard forms. \[ \int \frac{\dx}{a^2+x^2} = \frac1a\arctan\frac xa + C, \qquad \int\frac{\dx}{\sqrt{a^2-x^2}} = \arcsin\frac xa + C, \qquad \int \sec x \dx = \ln\abs{\sec x+\tan x} + C. \]
Repeated integration by parts
For \(\int P(x)g(x)\dx\) with \(P\) a polynomial, the tabular method applies integration by parts \(\deg P + 1\) times; see Useful tricks. If the original integral reappears after integrating by parts, as in \(\int e^x\sin x\dx\), the resulting equation can be solved for the integral.
Series
Convergence. The ratio test applies to terms built from factorials and powers, the root test to terms that are \(n\)th powers, comparison with a \(p\)-series to terms comparable to \(n^{-p}\), the alternating series test to alternating series with terms decreasing to \(0\), and the integral test to \(\sum f(n)\) with \(f\) positive and decreasing.
Value. Geometric series, telescoping series, and known power series evaluated at a point.
Limits
For indeterminate forms: L’Hôpital’s rule; Taylor expansion, when repeated applications of L’Hôpital’s rule do not simplify the quotient; the squeeze theorem. A recursively defined sequence that is monotone and bounded converges, and if \(a_{n+1}=g(a_n)\) with \(g\) continuous, its limit \(L\) satisfies \(L = g(L)\).
Multivariable integrals
Line integrals. If the field is conservative, \(F = \nabla\phi\), then \(\int_C F\cdot d\mathbf{r} = \phi(\text{end})-\phi(\text{start})\).
Closed curves in the plane. Green’s theorem.
Closed surfaces. The divergence theorem.
Other integrals. Parametrize the curve or surface, in coordinates adapted to the symmetry of the region.