Math, asked by elannick89, 1 month ago

integration of x^2/√x+5
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Answers

Answered by mathdude500
1

\large\underline{\sf{Solution-}}

Given integral is

\rm :\longmapsto\: \: I = \displaystyle \int \rm \: \dfrac{ {x}^{2} }{ \sqrt{x + 5} } dx

We use, here method of Substitution.

\red{\rm :\longmapsto\:Put \:  \sqrt{x + 5} = y}

\red{\rm :\longmapsto\:x + 5 =  {y}^{2}}

\red{\rm :\longmapsto\:dx  =  2{y}dy}

On substituting all these values in above integral, we get

\rm :\longmapsto\: \: I = \displaystyle \int \rm \: \dfrac{ {( {y}^{2}  - 5)}^{2} }{y}  \times 2ydy

\rm :\longmapsto\: \: I = 2\displaystyle \int \rm \: {( {y}^{2}  - 5)}^{2}dy

\rm :\longmapsto\: \: I = 2\displaystyle \int \rm \: {( {y}^{4} + 25 - 10 {y}^{2} )}dy

\rm :\longmapsto\: \: I = 2\bigg(\dfrac{ {y}^{5} }{5}  + 25y - \dfrac{10 {y}^{3} }{3} \bigg)  + c

 \:  \:  \:  \:  \:  \: \red{\bigg \{ \because \: \displaystyle \int \:  {x}^{n}dx = \dfrac{ {x}^{n + 1} }{n + 1}  + c  \bigg \}}

\rm :\longmapsto\: \: I = 2\bigg(\dfrac{ {( \sqrt{x + 5} ) \: }^{5} }{5}  + 25 \sqrt{x + 5}  - \dfrac{10 {( \sqrt{x + 5} ) \: }^{3} }{3} \bigg)  + c

Additional Information :-

\red{\rm :\longmapsto\:\displaystyle \int k\: dx \:  = kx \:  + c}

\red{\rm :\longmapsto\:\displaystyle \int  {e}^{x} \: dx \:  =  {e}^{x}  \:  + c}

\red{\rm :\longmapsto\:\displaystyle \int  \frac{1}{x} \: dx \:  =  log(x)  \:  + c}

\red{\rm :\longmapsto\:\displaystyle \int \: sinx \: dx \:  =  - \:  cosx \:  + c}

\red{\rm :\longmapsto\:\displaystyle \int \: cosx \: dx \:  =  \:  sinx \:  + c}

\red{\rm :\longmapsto\:\displaystyle \int \: cosecx \: dx \:  =  -  \:  cosecx  \: cotx\:  + c}

\red{\rm :\longmapsto\:\displaystyle \int \: secx \: dx \:  = \:  secx  \: tanx\:  + c}

\red{\rm :\longmapsto\:\displaystyle \int \:cotx \:  dx \:  =   log(sinx) \:  + c}

\red{\rm :\longmapsto\:\displaystyle \int \:tanx \:  dx \:  =   log(secx) \:  + c}

\red{\rm :\longmapsto\:\displaystyle \int \:secx \:  dx \:  =   log(secx + tanx) \:  + c}

\red{\rm :\longmapsto\:\displaystyle \int \:cosecx \:  dx \:  =   log(cosecx  -  cotx) \:  + c}

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