AHL5.15 extends the standard derivative table. Read it in reverse to obtain antiderivatives; do not assume that every derived integral below is a separate formula-booklet entry.
| Integrand | Antiderivative |
|---|---|
| \(a^x\) | \(\dfrac{a^x}{\ln a}+C\) |
| \(\sec x\tan x\) | \(\sec x+C\) |
| \(\operatorname{cosec}\,x\cot x\) | \(-\operatorname{cosec}\,x+C\) |
| \(\dfrac{1}{\sqrt{1-x^2}}\) | \(\arcsin x+C\) |
| \(-\dfrac{1}{\sqrt{1-x^2}}\) | \(\arccos x+C\) |
| \(\dfrac{1}{1+x^2}\) | \(\arctan x+C\) |
For the exponential form, \(a>0\) and \(a\ne1\). For the inverse-sine and inverse-cosine forms, \(-1<x<1\). Trigonometric calculus uses radians.
Scaled inverse-trigonometric forms
\(\int\dfrac{1}{a^2+x^2}\,\mathrm{d}x=\dfrac{1}{a}\arctan\left(\dfrac{x}{a}\right)+C\qquad(a>0)\)
\(\int\dfrac{1}{\sqrt{a^2-x^2}}\,\mathrm{d}x=\arcsin\left(\dfrac{x}{a}\right)+C\qquad(a>0,\ |x|<a)\)
Complete the square and scale the variable before applying these forms. For a difference of squares in a rational denominator, derive logarithms using partial fractions.
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