Calculus/Tables of Integrals

Rules

• $\int cf(x)\,dx = c\int f(x)\,dx$
• $\int f(x)+g(x)\,dx = \int f(x)\,dx+ \int g(x)\,dx$
• $\int f(x)-g(x)\,dx = \int f(x)\,dx- \int g(x)\,dx$
• $\int u\,dv\ = uv - \int v\,du$

Powers

• $\int dx = x+C$
• $\int a\,dx = ax+C$
• $\int x^n\,dx = \frac{1}{n+1}x^{n+1}+C\qquad\mbox{ if }n\ne-1$
• $\int {1\over x}\,dx = \ln|x|+C$
• $\int \frac{1}{ax+b}\,dx = {1 \over a}\ln|ax+b|+C\qquad\mbox{ if }a\ne 0$

Trigonometric Functions

Basic Trigonometric Functions

• $\int \sin{x}\,dx = -\cos{x} + C$
• $\int \cos{x}\,dx = \sin{x} + C$
• $\int \tan{x}\,dx = \ln \left |{\sec{x}} \right | + C$
• $\int \sin^2{x}\,dx = \tfrac{1}{2} x - \tfrac{1}{4} \sin{2x} + C$
• $\int \cos^2{x}\,dx = \tfrac{1}{2} x + \tfrac{1}{4} \sin{2x} + C$
• $\int \tan^2{x}\,dx = \tan(x) - x + C$
• $\int\sin^n x\,dx = -\frac{\sin^{n-1} x\cos x}{n} + \frac{n-1}{n}\int\sin^{n-2} x\,dx+C \qquad\mbox{(for }n>0\mbox{)}$

• $\int\cos^n x\,dx = -\frac{\cos^{n-1} x\sin x}{n} + \frac{n-1}{n}\int\cos^{n-2} x\,dx+C \qquad\mbox{(for }n>0\mbox{)}\,\!$

• $\int\tan^n x\,dx = \frac{1}{(n-1)}\tan^{n-1} x-\int\tan^{n-2} x\,dx+C \qquad\mbox{(for }n\neq 1\mbox{)}$

Reciprocal Trigonometric Functions

• $\int \sec{x}\,dx = \ln \left |{\sec{x}}+\tan x \right | + C = \ln \left | \tan{\left( \frac1 2 x +\frac1 4 \pi \right) }\right |+C$
• $\int \csc{x}\,dx = - \ln \left |{\csc x + \cot x} \right | + C=\ln \left | \tan \left(\frac1 2 x \right) \right |+C$
• $\int \cot{x}\,dx = \ln \left |{\sin{x}} \right | + C$

• $\int \sec^2 kx\,dx = \frac1 k \tan{kx} + C$
• $\int \csc^2 kx\,dx = -\frac1 k \cot kx + C$
• $\int \cot^2 kx\,dx = -x-\frac1 k \cot kx + C$
• $\int \sec{x}\tan{x}\,dx = \sec x + C$
• $\int \sec x \csc x\,dx =\ln \left | \tan x \right | + C$

• $\int \sec^n{x}\,dx = \frac{\sec^{n-1}{x} \sin {x}}{n-1} + \frac{n-2}{n-1}\int \sec^{n-2}{x}\,dx+C \qquad \mbox{ (for }n \ne 1\mbox{)}$

• $\int \csc^n{x}\,dx = -\frac{\csc^{n-1}{x} \cos{x}}{n-1} + \frac{n-2}{n-1}\int \csc^{n-2}{x}\,dx+C \qquad \mbox{ (for }n \ne 1\mbox{)}$

• $\int\cot^n x\,dx = -\frac{1}{n-1}\cot^{n-1} x - \int\cot^{n-2} x\,dx+C \qquad\mbox{(for }n\neq 1\mbox{)}$

Inverse Trigonometric Functions

• $\int {1\over \sqrt{1-x^2}}\,dx = \mbox{arcsin}(x) + C$
• $\int {1\over \sqrt{a^2-x^2}}\,dx = \mbox{arcsin}(x/a) + C \qquad\mbox{ if }a\ne 0$
• $\int {1\over 1+x^2}\,dx = \mbox{arctan}(x) + C$
• $\int {1\over a^2+x^2}\,dx = {1\over a}\mbox{arctan}(x/a) + C \qquad\mbox{ if }a\ne 0$

Exponential and Logarithmic Functions

• $\int e^x \,dx = e^x + C$
• $\int e^{ax} \,dx = {1\over a}e^{ax} + C \qquad\mbox{ if }a\neq 0$
• $\int a^x \,dx = {1\over \ln a}a^x + C \qquad\mbox{ if }a>0, a\neq 1$
• $\int \ln x \,dx = x\ln x-x + C$

Inverse Trigonometric Functions

• $\int \mbox{arcsin}(x) \,dx = x\,\mbox{arcsin}(x) + \sqrt{1-x^2} + C$
• $\int \mbox{arccos}(x) \,dx = x\,\mbox{arccos}(x) - \sqrt{1-x^2} + C$
• $\int \mbox{arctan}(x) \,dx = x\,\mbox{arctan}(x) - {1\over 2}\ln(1+x^2) + C$