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Eight complete paths: when to choose the method, a worked example and three challenges to try yourself.

Substitution →Cyclic integration by parts →Repeated linear factors →Repeated quadratics and mixed factors →Changing integration order and Fubini →Polar and cylindrical coordinates →Cylindrical coordinates →Spherical coordinates →
1Essentials

Powers and linearity

∫xn dx=xn+1n+1+C,n≠−1\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\ne-1View explanation2Recognizing patterns

Substitution

u=g(x),du=g′(x) dxu=g(x),\quad du=g^{\prime}(x)\,dxView explanation3Products

Integration by parts

∫u dv=uv−∫v du\int u\,dv=uv-\int v\,duView explanation4Rational functions

Partial fractions

1x2−1=12(x−1)−12(x+1)\frac{1}{x^2-1}=\frac{1}{2(x-1)}-\frac{1}{2(x+1)}View explanation5Identities

Trigonometric integrals

sin⁡2x=1−cos⁡(2x)2\sin^2x=\frac{1-\cos(2x)}{2}View explanation6Regions and order

Double and triple integrals

∫cd∫a(y)b(y)f(x,y) dx dy\int_c^d\int_{a(y)}^{b(y)} f(x,y)\,dx\,dyView explanation7Circular symmetry

Polar and cylindrical coordinates

x=rcos⁡θ, y=rsin⁡θ,dA=r dr dθx=r\cos\theta,\ y=r\sin\theta,\quad dA=r\,dr\,d\thetaView explanation8Spatial symmetry

Spherical coordinates

dV=r2sin⁡ϕ dr dϕ dθdV=r^2\sin\phi\,dr\,d\phi\,d\thetaView explanation9Limits

Improper integrals and convergence

∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx\int_a^{\infty}f(x)\,dx=\lim_{b\to\infty}\int_a^b f(x)\,dxView explanation10Advanced techniques

Beta and gamma functions

Γ(s)=∫0∞ts−1e−t dt,s>0\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}\,dt,\quad s>0View explanation11Advanced techniques

Green, Stokes and the divergence theorem

∮∂DP dx+Q dy=∬D(Qx−Py) dA\oint_{\partial D}P\,dx+Q\,dy=\iint_D(Q_x-P_y)\,dAView explanation12Rational functions

Proper fraction: lower numerator degree

deg⁡P<deg⁡Q\deg P<\deg QView explanation13Rational functions

Equal or higher degree: divide first

PQ=S+RQ,deg⁡R<deg⁡Q\frac{P}{Q}=S+\frac{R}{Q},\quad\deg R<\deg QView explanation14Rational functions

Distinct linear factors

P(x−a)(x−b)=Ax−a+Bx−b\frac{P}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}View explanation15Rational functions

Repeated linear factors

A1x−a+A2(x−a)2+⋯+Am(x−a)m\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots+\frac{A_m}{(x-a)^m}View explanation16Rational functions

Irreducible quadratic factors

Ax+Bax2+bx+c,b2−4ac<0\frac{Ax+B}{ax^2+bx+c},\quad b^2-4ac<0View explanation17Rational functions

Repeated quadratics and mixed factors

∑k=1mAkx+Bk(ax2+bx+c)k\sum_{k=1}^m\frac{A_kx+B_k}{(ax^2+bx+c)^k}View explanation18Integration techniques

Direct primitives and the chain rule

∫f′(x)g(f(x)) dx=G(f(x))+C\int f'(x)g(f(x))\,dx=G(f(x))+CView explanation19Integration techniques

The logarithmic f′/f pattern

∫f′(x)f(x) dx=ln⁡∣f(x)∣+C\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+CView explanation20Rational functions

Completing the square

ax2+bx+c=a(x+b2a)2+c−b24aax^2+bx+c=a\left(x+\frac b{2a}\right)^2+c-\frac{b^2}{4a}View explanation21Integration techniques

Repeated and tabular integration by parts

∫P(x)eax dx\int P(x)e^{ax}\,dxView explanation22Integration techniques

Cyclic integration by parts

I=A−kII=A-kIView explanation23Integration techniques

Reduction formulas

In=∫sin⁡nx dxI_n=\int\sin^n x\,dxView explanation24Integration techniques

Sine and cosine: odd and even powers

∫sin⁡mxcos⁡nx dx\int\sin^m x\cos^n x\,dxView explanation25Integration techniques

Tangent, secant, cotangent and cosecant powers

1+tan⁡2x=sec⁡2x1+\tan^2x=\sec^2xView explanation26Integration techniques

Product-to-sum and double-angle identities

sin⁡acos⁡b=12[sin⁡(a+b)+sin⁡(a−b)]\sin a\cos b=\tfrac12[\sin(a+b)+\sin(a-b)]View explanation27Integration techniques

Trigonometric substitution: three radical forms

a2−x2,a2+x2,x2−a2\sqrt{a^2-x^2},\quad\sqrt{a^2+x^2},\quad\sqrt{x^2-a^2}View explanation28Integration techniques

The t = tan(x/2) substitution

sin⁡x=2t1+t2,cos⁡x=1−t21+t2\sin x=\frac{2t}{1+t^2},\quad\cos x=\frac{1-t^2}{1+t^2}View explanation29Integration techniques

Hyperbolic substitutions

cosh⁡2t−sinh⁡2t=1\cosh^2t-\sinh^2t=1View explanation30Integration techniques

Rationalizing radicals and fractional powers

x=tmx=t^mView explanation31Advanced techniques

Euler substitutions for quadratic radicals

ax2+bx+c=t+a x(a>0)\sqrt{ax^2+bx+c}=t+\sqrt a\,x\quad(a>0)View explanation32Integration techniques

Exponential and logarithmic substitutions

t=ex,dx=dt/tt=e^x,\quad dx=dt/tView explanation33Integration techniques

Definite integrals and the fundamental theorem

∫abf(x) dx=F(b)−F(a)\int_a^bf(x)\,dx=F(b)-F(a)View explanation34Integration techniques

Symmetry, parity and periodicity

∫−aafimpar(x) dx=0\int_{-a}^af_{\mathrm{impar}}(x)\,dx=0View explanation35Integration techniques

Absolute values and piecewise functions

∫abf=∑j∫cjcj+1fj\int_a^bf=\sum_j\int_{c_j}^{c_{j+1}}f_jView explanation36Advanced techniques

Trapezoidal and Simpson rules

Sn=h3[f0+4f1+2f2+⋯+4fn−1+fn]S_n=\frac h3[f_0+4f_1+2f_2+\cdots+4f_{n-1}+f_n]View explanation37Multiple variables

Changing integration order and Fubini

∬Df dA\iint_Df\,dAView explanation38Multiple variables

General variable changes and the Jacobian

dA=∣det⁡∂(x,y)∂(u,v)∣du dvdA=\left|\det\frac{\partial(x,y)}{\partial(u,v)}\right|du\,dvView explanation39Multiple variables

Cylindrical coordinates

dV=r dr dθ dzdV=r\,dr\,d\theta\,dzView explanation40Advanced techniques

Differentiation with respect to a parameter

dda∫ABf(x,a) dx=∫AB∂af(x,a) dx\frac d{da}\int_A^B f(x,a)\,dx=\int_A^B\partial_af(x,a)\,dxView explanation41Advanced techniques

Integration using series

∫∑n=0∞anxn dx=∑n=0∞anxn+1n+1+C\int\sum_{n=0}^\infty a_nx^n\,dx=\sum_{n=0}^\infty\frac{a_nx^{n+1}}{n+1}+CView explanation42Advanced techniques

Nonelementary primitives and special functions

∫e−x2 dx=π2erf⁡(x)+C\int e^{-x^2}\,dx=\frac{\sqrt\pi}{2}\operatorname{erf}(x)+CView explanation43Advanced techniques

Complex contours and residues

∮γf(z) dz=2πi∑Res⁡(f)\oint_\gamma f(z)\,dz=2\pi i\sum\operatorname{Res}(f)View explanation