# Antiderivatives (Standard integrals)

﻿Standard integrals (I)

${\int }^{\frac{\mathrm{d}\mathrm{x}}{{\mathrm{a}}^{2}-{\mathrm{x}}^{2}\phantom{\rule{mediummathspace}{0ex}}}}=\frac{1}{2\mathrm{a}}\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{\mathrm{a}+\mathrm{x}}{\mathrm{a}-\mathrm{x}}\right)$ + C

${\int }^{\frac{\mathrm{d}\mathrm{x}}{{\mathrm{x}}^{2}-{\mathrm{a}}^{2}\phantom{\rule{mediummathspace}{0ex}}}}=\frac{1}{2\mathrm{a}}\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{\mathrm{x}-\mathrm{a}}{\mathrm{a}+\phantom{\rule{mediummathspace}{0ex}}\mathrm{x}}\right)$ + C

${\int }^{\frac{\mathrm{d}\mathrm{x}}{{\mathrm{x}}^{2}+\phantom{\rule{mediummathspace}{0ex}}{\mathrm{a}}^{2}\phantom{\rule{mediummathspace}{0ex}}}}=\frac{1}{\mathrm{a}}{\mathrm{tan}}^{-1}\frac{\mathrm{x}}{\mathrm{a}}$ + C

${\int }^{\frac{\mathrm{d}\mathrm{x}}{\sqrt{{\mathrm{a}}^{2}-{\mathrm{x}}^{2}\phantom{\rule{mediummathspace}{0ex}}}\phantom{\rule{mediummathspace}{0ex}}}}={\mathrm{sin}}^{-1}\frac{\mathrm{x}}{\mathrm{a}}$ + C

${\int }^{\frac{\mathrm{d}\mathrm{x}}{\sqrt{{\mathrm{x}}^{2}-{\mathrm{a}}^{2}\phantom{\rule{mediummathspace}{0ex}}}\phantom{\rule{mediummathspace}{0ex}}}}=\mathrm{log}\phantom{\rule{mediummathspace}{0ex}}\left(\phantom{\rule{mediummathspace}{0ex}}\mathrm{x}+\sqrt{{\mathrm{x}}^{2}-{\mathrm{a}}^{2}\phantom{\rule{mediummathspace}{0ex}}}\right)\phantom{\rule{mediummathspace}{0ex}}$+ C

${\int }^{\frac{\mathrm{d}\mathrm{x}}{\sqrt{{\mathrm{x}}^{2}+\phantom{\rule{mediummathspace}{0ex}}{\mathrm{a}}^{2}\phantom{\rule{mediummathspace}{0ex}}}\phantom{\rule{mediummathspace}{0ex}}}}=\mathrm{log}\phantom{\rule{mediummathspace}{0ex}}\left(\phantom{\rule{mediummathspace}{0ex}}\mathrm{x}+\sqrt{{\mathrm{x}}^{2}+{\mathrm{a}}^{2}\phantom{\rule{mediummathspace}{0ex}}}\right)\phantom{\rule{mediummathspace}{0ex}}$ + C

Standard integrals (II)

Formula for Integration by parts

∫ (uv) dx = u∫ vdx- ∫ ($\frac{\mathrm{d}\mathrm{u}}{\mathrm{d}\mathrm{x}}\right)\phantom{\rule{mediummathspace}{0ex}}{\int }^{\mathrm{v}\mathrm{d}\mathrm{x}\phantom{\rule{mediummathspace}{0ex}}}$

∫ eax cosbxdx=eax $\frac{\left(\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{b}\mathrm{x}\right)}{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}}$

∫ eax sinbxdx=eax

Standard integral of trigonometrical functions:

Integral of hyperbolic function:

∫sinhx dx = coshx+ c

∫ coshx dx = sinhx+ c

∫tanhx dx = ln(coshx )+ c

∫ cotdx = ln|sinhx|+ c

∫ sechx dx= tan-1 |sinhx|+c

∫ cosechx dx= ln |tan$\frac{\mathrm{x}}{2}$ |+c

﻿Standard integral (III)

$\frac{\mathrm{d}\mathrm{x}}{\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{x}+\mathrm{b}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{x}}$

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