INTEGRATION AS INVERSE PROCESS OF DIFFERENTIATION

Integral means anti – derivative of a function. If \frac{d}{{dx}}\left( {f\left( x \right)} \right) = F\left( x \right), then f\left( x \right)is anti – derivative of F\left( x \right) or \int {F\left( x \right)} \,dx = f\left( x \right).

\int {F\left( x \right)} \,dxwill have infinite number of values, so it is called indefinite integral of f\left( x \right).

\frac{d}{{dx}}\left( K \right) = 0{Derivative of any constant is zero}

\therefore \frac{d}{{dx}}\left( {2x} \right) = 2

\frac{d}{{dx}}\left( {2x + 1} \right) = 2

\frac{d}{{dx}}\left( {2x + 5} \right) = 2

\frac{d}{{dx}}\left( {2x - 3} \right) = 2

\therefore \int {F\left( x \right)} \,dx = f\left( x \right) + c . 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( x \right)\, = \,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\int {1\,dx\,\, = \,\,\int {{x^0}\,dx} \,\, = \,\,x + c\,} \,
  3. \frac{d}{{dx}}\left( {\log \left( x \right)} \right) = \frac{1}{x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\int {\frac{1}{x}dx} \, = \,\log \left| x \right|\, + \,c
  4. \frac{d}{{dx}}\left( {{e^x}} \right) = {e^x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\int {{e^x}\,dx\,\, = \,{e^x}\, + \,c}
  5. \frac{d}{{dx}}\left( {{a^x}} \right)\, = \,{a^x}\,{\log _e}a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\int {{a^x}dx\,\, = \,\frac{{{a^x}}}{{{{\log }_e}a}},\,\,\,\,a > 0}
  6. \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}
  7. \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}
  8. \frac{d}{{dx}}\left( {\sqrt x } \right) = \,\frac{1}{{2\sqrt x }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\int {\frac{1}{{\sqrt x }}\,dx\, = \,2\sqrt x } \, + c
  9. \frac{d}{{dx}}\left( {\sin x} \right)\,\, = \,\cos x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\int {\cos x\,dx\, = \,\sin x}
  10. \frac{d}{{dx}}\left( {\cos x} \right)\, = \, - \sin x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\int {\sin x\,dx\,\, = \,\cos x\, + \,c}
  11. \frac{d}{{dx}}\left( {\tan x} \right)\, = \,{\sec ^2}x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\int {{{\sec }^2}xd\,x\, = \,\tan x + c} \,
  12. \frac{d}{{dx}}\left( {\cot x} \right)\, = \, - \cos e{c^2}x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \int {\cos {e^2}x\,dx\, = \, - \cot x\, + c}
  13. \frac{d}{{dx}}\left( {\sec x} \right)\, = \,\sec x\,\tan x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \int {\sec x\,\tan x\,dx\, = \,\sec x\, + c}
  14. \frac{d}{{dx}}\left( {\cos ecx} \right)\, = \, - \cos ecx\,\cot \,x\,\,\, \Rightarrow \,\,\int {\cos ecx\,\,\cot x\,\,dx\, = \, - \cos ecx\, + c}
  15. \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}
  16. \frac{d}{{dx}}\left( {{{\cos }^{ - 1}}x} \right)\, = \, - \frac{1}{{\sqrt {1 - {x^2}} }}\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \int {\frac{1}{{\sqrt {1 - {x^2}} }}\,dx\, = \, - {{\cos }^{ - 1}}x\, + c}
  17. \frac{d}{{dx}}\left( {{{\tan }^{ - 1}}x} \right)\, = \,\frac{1}{{1 + {x^2}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \int {\frac{1}{{1 + {x^2}}}\,dx\, = \,{{\tan }^{ - 1}}x\, + c}
  18. \frac{d}{{dx}}\left( {{{\cot }^{ - 1}}x} \right)\, = \, - \frac{1}{{1 + {x^2}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \int {\frac{1}{{1 + {x^2}}}dx\, = \, - {{\cot }^{ - 1}}x\, + c}
  19. \frac{d}{{dx}}\left( {{{\sec }^{ - 1}}x} \right)\, = \,\frac{1}{{x\sqrt {{x^2} - 1} }}\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \int {\frac{1}{{x\sqrt {{x^2} - 1} }}\,dx\, = \,{{\sec }^{ - 1}}x\, + c}
  20. \frac{d}{{dx}}\left( {\cos e{c^{ - 1}}x} \right)\, = \, - \frac{1}{{x\sqrt {{x^2} - 1} }}\,\,\,\, \Rightarrow \int {\frac{1}{{x\sqrt {{x^2} - 1} }}\,dx\, = \,\cos e{c^{ - 1}}x\, + c}

\int {K\,f\left( x \right)dx\,\, = \,\,K\int {f\left( x \right)} \,dx}

\int {\left[ {f\left( x \right) + g\left( x \right)} \right]} \,dx\, = \,\int {f\left( x \right)} \,dx\, + \,\int {g\left( x \right)} \,dx

\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

 

Post Author: E-Maths