Methods of Integration> Integration by Parts

\[\int\left(\mathrm{u.v}\right)\mathrm{dx}=\mathrm{u}\int\mathrm{v}\mathrm{dx}-\int\left(\frac{\mathrm{du}}{\mathrm{dx}}\right).\left(\int\mathrm{v}\mathrm{dx}\right)\mathrm{dx}\]

Special Result:

∫ eˣ [f(x) + f′(x)] dx = eˣ f(x) + C

First function should be chosen in the following order of preference:

L → Logarithmic function
I → Inverse trigonometric function
A → Algebraic function
T → Trigonometric function
E → Exponential function

Note:

For the integration of logarithmic or inverse trigonometric functions alone, take unity (1) as the second function.

Standard forms:

i) \[\int\sqrt{x^{2}+a^{2}}dx=\frac{1}{2}\left[ \begin{array} {c}{x\sqrt{x^{2}+a^{2}}} {+a^{2}\log|x+\sqrt{x^{2}+a^{2}|}} \end{array}\right]+C\]

ii) \[\int\sqrt{a^{2}-x^{2}}dx=\frac{1}{2}\left[x\sqrt{a^{2}-x^{2}}+a^{2}\sin^{-1}\left(\frac{x}{a}\right)\right]+C\]

iii) \[\int\sqrt{x^{2}-a^{2}}dx=\frac{1}{2}[x\sqrt{x^{2}-a^{2}}-a^{2}\log|x+\sqrt{x^{2}-a^{2}}|]\] + C