formula – of – indefinite – integrals

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Formula Of  Indefinite Integrals

(i) \dfrac{d}{dx} \dfrac{x^{n+1}}{n+1}  =  x^n , n \ne -1

\Rightarrow  \int x^n d x = \dfrac{x^{n+1}}{n+1} + C, n \ne -1 

(ii) \dfrac{d}{dx}(log | x | ) = \dfrac{1}{x} 

\Rightarrow  \int \dfrac{1}{x} d x  =  log  | x | + C , when x \ne 0

(iii) \dfrac{d}{dx} (e^x)  =  e^e 

\Rightarrow \int e^x d x  = e^x +  C

(iv) \dfrac{d}{dx} (\dfrac{1}{a}  cosec^{-1}  \dfrac{x}{a})  =  \dfrac{-1}{x \sqrt{x^2 –  a^2}}

\Rightarrow  \int \dfrac{-dx}{x\sqrt{x^2 – a^2}} = \dfrac{1}{a} cosec^{-1} (\dfrac{x}{a}) + C

(v) \dfrac{d}{dx} (\dfrac{1}{a}sec^{-1} \dfrac{x}{a}) = \dfrac{1}{x \sqrt{x^2  –  a^2}}

\Rightarrow \int \dfrac{dx}{x \sqrt{x^2 – a^2}} = \dfrac{1}{a} sec^{-1} (\dfrac{x}{a}) + C

(vi) \dfrac{d}{dx} (\dfrac{1}{a} cot^{-1} \dfrac{x}{a} ) = \dfrac{1}{a^2 + x^2}

  \Rightarrow \int \dfrac{-1}{a^2 + b^2} d x = \dfrac{1}{a} cot^{-1} \dfrac{x}{a} + C

(vii) \dfrac{d}{dx} (\dfrac{1}{a} tan^{-1} \dfrac{x}{a}) = \dfrac{1}{a^2 + b^2}

\Rightarrow \int \dfrac{dx}{a^2  + x^2 } = \dfrac{1}{a} tan^{-1} (\dfrac{x}{a}) + C

(viii)  \dfrac{d}{dx} (cos^{-1} \dfrac{x}{a}) = \dfrac{-1}{\sqrt{a^2 – x^2}} 

\Rightarrow \int \dfrac{-1}{\sqrt{a^2 – x^2}} d x  = cos^1  (\dfrac{x}{a}) + C

(ix) \dfrac{d}{dx} (sin^{-1} \dfrac{x}{a}) = \dfrac{1}{\sqrt{a^2 – x^2}}

\Rightarrow \int \dfrac{dx}{\sqrt{a^2  –  x^2}}  = sin^1 (\dfrac{x}{a})  +  C

(x) \dfrac{d}{dx} (log | cosec x – cot x | ) = cosec x

\Rightarrow \int cosec  x dx = log | cosec  x – cot  x | + C

(xi) \frac{d}{d x}  (-log|sin x | ) = cot x

\Rightarrow  \int cot  xdx  =  log | sin  x |  +  C 

(xii) \frac{d}{dx} (- log | cos  x  |)   =  tan x

\Rightarrow  \int tan  xdx  = –  log | cos  x |  +  C 

(xiii) \frac{d}{dx} ( log | sec  x  +  tan  x|)   =  sec x

\Rightarrow  \int sec  xdx  =  log | sec  x  +  tan  x|  +  C 

(xiv) \frac{d}{dx} ( log | cosec  x  –  cot  x|)   =  cosec x

\Rightarrow  \int cosec  xdx  =  log | cosec  x  –  cot  x|  +  C 

(xv) \frac{d}{dx} ( \sin^{-1} (\frac{x}{a})) = \frac{1}{\sqrt{a^2  –  x^2}}

\Rightarrow  \int \frac{dx}{\sqrt{a^2  –  x^2}}  =  sin^{-1} (\frac{x}{a})  +  C 

(xvi) \frac{d}{dx} ( \cos^{-1} (\frac{x}{a})) = \frac{-1}{\sqrt{a^2  –  x^2}}

\Rightarrow  \int \frac{-1}{\sqrt{a^2  –  x^2}} dx  =  cos^{-1} (\frac{x}{a})  +  C 

(xvii) \frac{d}{dx}  (\frac{1}{a} \tan^{-1} (\frac{x}{a}))  =  \frac{1}{a^2  +  x^2}

\Rightarrow  \int \frac{dx}{a^2  +  x^2}  = \frac{1}{a} tan^{-1} (\frac{x}{a})  +  C 

(xviii) \frac{d}{dx}  (\frac{1}{a} \cot^{-1} (\frac{x}{a}))  =  \frac{-1}{a^2  +  x^2}

\Rightarrow  \int \frac{-1}{a^2  +  x^2} dx  = \frac{1}{a} \cot^{-1} (\frac{x}{a})  +  C 

(xix) \frac{d}{dx}  (\frac{1}{a} \sec^{-1} (\frac{x}{a}))  =  \frac{1}{x \sqrt{x^2  –  a^2}}

\Rightarrow  \int \frac{dx}{x \sqrt{x^2  –  a^2}}  = \frac{1}{a} \sec^{-1} (\frac{x}{a})  +  C 

(xx)

Formula Of Indefinite Integrals

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