Place keep common integrals for lookup. Constant of integration is assumed \(c\) and not shown to save space.
| Integral | anti derivative |
| \(\int \cos x\ dx\) | \(\sin x\) |
| \(\int \sin x\ dx\) | \(-\cos x\) |
| \(\int \tan x\ dx\) | \(-\ln \left ( \cos x\right ) \) |
| \(\int \frac {1}{x^{2}+a}\ dx\) | \(\frac {1}{\sqrt {a}}\arctan \left ( \frac {x}{\sqrt {a}}\right ) \) |
| \(\int \frac {1}{x^{2}-a}\ dx\) | \(-\frac {1}{\sqrt {a}}\operatorname {arctanh}\left ( \frac {x}{\sqrt {a}}\right ) \) |
| \(\int \frac {1}{\sqrt {-x^{2}+a}}\ dx\) | \(\arcsin \left ( \frac {x}{\sqrt {a}}\right ) \) |
| \(\int \frac {1}{\sqrt {x^{2}+a^{2}}}\ dx\) | \(\operatorname {arcsinh}\left ( \frac {x}{\sqrt {a}}\right ) \) |
| \(\int \frac {1}{\cos x}dx=\int \sec xdx\) | \(\ln \left ( \sec x+\tan x\right ) \) |
| \(\int \frac {1}{\sin x}dx=\int \csc xdx\) | \(\ln \left ( \csc x-\cot x\right ) \) |
Integration by part is \(\int udv=uv-\int vdu\). For \(u\) pick the one which when differentiating it repeatedly it vanishes. So given integrand say \(x^{2}\sin x\), use \(u=x^{2}\) and \(dv=\sin x\).
For partial fractions uses \(\frac {f\left ( x\right ) }{\left ( {}\right ) \left ( {}\right ) }=\frac {A}{\left ( {}\right ) }+\frac {B}{\left ( {}\right ) }\) and so on. Then find \(A,B\) by setting \(x\) to different values.