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# INTEGRATION AS INVERSE PROCESS OF DIFFERENTIATION

Home » Integration » INTEGRATION AS INVERSE PROCESS OF DIFFERENTIATION

Integral means anti – derivative of a function. If , then is anti – derivative of or .

will have infinite number of values, so it is called indefinite integral of .

{Derivative of any constant is zero}

. where c is any constant called arbitrary or integration constant. We must write +c in every indefinite integrals.

Integrals of some important functions are given below:

1. $$\frac{d}{dx}\left( \frac{{{x}^{n+1}}}{n+1} \right)={{x}^{n}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\int{{{x}^{n}}dx}=\frac{{{x}^{n+1}}}{n+1}+c,\,\,n\ne -1$$
2. $$\frac{d}{dx}\left( \frac{{{x}^{-n+1}}}{-n+1} \right)\,=\,\frac{1}{{{x}^{n}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\int{\frac{1}{{{x}^{n}}}\,dx\,=\,\frac{{{x}^{-n+1}}}{-n+1}\,+c}$$
3. $$\frac{d}{dx}\left( \frac{2}{3}{{x}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}} \right)=\sqrt{x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,\int{\sqrt{x}\,dx\,\,=\,\frac{2}{3}{{x}^{{}^{3}\!\!\diagup\!\!{}_{2}\;}}\,+\,c}$$
4. $$\displaystyle \frac{d}{dx}\left( {{\sin }^{-1}}x \right)\,=\,\frac{1}{\sqrt{1-{{x}^{2}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \int{\frac{1}{\sqrt{1-{{x}^{2}}}}\,dx\,=\,-{{\sin }^{-1}}x\,+c}$$

$$\int{\left[ {{f}_{1}}\left( x \right)\pm {{f}_{2}}\left( x \right)\pm {{f}_{3}}\left( x \right)\pm ……..\pm {{f}_{n}}\left( x \right) \right]}=\int{{{f}_{1}}\left( x \right)\,dx\pm \int{{{f}_{2}}\left( x \right)dx}\pm \int{{{f}_{3}}\left( x \right)}\,dx}\pm …….\pm \int{{{f}_{n}}\left( x \right)}\,dx$$

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