Integration reverses differentiation. An indefinite integral represents a family of functions. f (x), dx=f(x)+C Example Differentiating f(x)=x3+2 gives f (x)=3x2. 3x2,dx=x3+C The constant C is arbitrary because every constant differentiates to zero. Formula overview Power functions: xn,dx= xn+1n+1+C for n!= -1, and (ax+b)n,dx= (ax+b)n+1a(n+1)+C for a 0 and n!= -1. Reciprocal functions: 1x,dx=ln|x|+C, and 1ax+b,dx= 1aln|ax+b|+C for a 0. Exponential functions: ex,dx=ex+C, and eax+b,dx= 1a eax+b+C for a 0. Trigonometric functions: x,dx=sin x+C, x,dx=-cos x+C, and 2x,dx=tan x+C. For a linear argument, (ax+b),dx= sin(ax+b)a+C, (ax+b),dx=- cos(ax+b)a+C, and 2(ax+b),dx= tan(ax+b)a+C, where a 0.

Integration reverses differentiation. An indefinite integral represents a family of functions.

\(\int \mathrm{f}^{\prime}(x)\,\mathrm dx=\mathrm{f}(x)+C\)

Differentiating \(\mathrm{f}(x)=x^3+2\) gives \(\mathrm{f}^{\prime}(x)=3x^2\).

The constant \(C\) is arbitrary because every constant differentiates to zero.

Power functions: \(\displaystyle\int x^n\,\mathrm{d}x=\dfrac{x^{n+1}}{n+1}+C\) for \(n\ne-1\), and \(\displaystyle\int (ax+b)^n\,\mathrm{d}x=\dfrac{(ax+b)^{n+1}}{a(n+1)}+C\) for \(a\ne0\) and \(n\ne-1\).

Reciprocal functions: \(\displaystyle\int \dfrac1x\,\mathrm{d}x=\ln|x|+C\), and \(\displaystyle\int \dfrac1{ax+b}\,\mathrm{d}x=\dfrac1a\ln|ax+b|+C\) for \(a\ne0\).

Exponential functions: \(\displaystyle\int e^x\,\mathrm{d}x=e^x+C\), and \(\displaystyle\int e^{ax+b}\,\mathrm{d}x=\dfrac1a e^{ax+b}+C\) for \(a\ne0\).

Trigonometric functions: \(\displaystyle\int\cos x\,\mathrm{d}x=\sin x+C\), \(\displaystyle\int\sin x\,\mathrm{d}x=-\cos x+C\), and \(\displaystyle\int\sec^2x\,\mathrm{d}x=\tan x+C\).

For a linear argument, \(\displaystyle\int\cos(ax+b)\,\mathrm{d}x=\dfrac{\sin(ax+b)}a+C\), \(\displaystyle\int\sin(ax+b)\,\mathrm{d}x=-\dfrac{\cos(ax+b)}a+C\), and \(\displaystyle\int\sec^2(ax+b)\,\mathrm{d}x=\dfrac{\tan(ax+b)}a+C\), where \(a\ne0\).

Basics of Integration

Basics of Integration