Basics of Integration

IB AA SL: Core and Optional Content

Core: power, exponential, logarithmic, sine and cosine antiderivatives. Optional: formulae that assume sec notation.

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

Example

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

\(\int3x^2\,\mathrm{d}x=x^3+C\)

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

Formula overview

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 \(a\ne0\).

For squared trigonometric functions, use \(\cos^2x=\dfrac{1+\cos2x}{2}\), \(\sin^2x=\dfrac{1-\cos2x}{2}\), and \(\tan^2x=\sec^2x-1\) before integrating.