Inverse Functions

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Inverse Functions The inverse of any function $ displaystyle (f)$ is a function that will do the opposite of $ displaystyle f$. In other words the function that will undo the effects of $ displaystyle f$. So, if $ displaystyle f$ maps 4 into 13, then the inverse of $ displaystyle f$ will map 13 […]

Integration of Inverse Trigonometric Forms

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Integration of a inverse Trigtonometric Forms We know the derivatives of the inverse trigonometric functions $displaystyle (text{arcsinx}{)}’=frac{1}{{sqrt{{1-{{x}^{2}}}}}}$ $displaystyle (text{arccosx}{)}’=frac{{-1}}{{sqrt{{1-{{x}^{2}}}}}}$ $displaystyle (arctgx{)}’=frac{1}{{1+{{x}^{2}}}}$ $displaystyle (text{arccotgx}{)}’=frac{{-1}}{{1+{{x}^{2}}}}$ $displaystyle (text{arcsecx}{)}’=frac{1}{{left| x right|sqrt{{{{x}^{2}}-1}}}}$ $displaystyle (text{arccscx}{)}’=frac{{-1}}{{left| x right|sqrt{{{{x}^{2}}-1}}}}$ Using those derivatives above, we can obtain the integrals as below, where u is a function of x that u=f(x). $ displaystyle int{{frac{{du}}{{sqrt{{{{a}^{2}}-{{u}^{2}}}}}}}}=arcsin frac{u}{a}+C$ $ displaystyle int{{frac{{du}}{{{{a}^{2}}+{{u}^{2}}}}}}=frac{1}{a}arctan frac{u}{a}+C$ $displaystyle […]