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  • The integral <img alt="\int \frac{dx}{(x+4)^{8/7}(x-3)^{6/7}}" src="https://learn.careers360.com/latex-image/?%5Cint%20%5Cfrac%7Bdx%7D%7B%28x&plus;4%29%5E%7B8/7%7D%28x-3%29%5E%7B6/7%7D%7D"

The integral \int \frac{dx}{(x+4)^{8/7}(x-3)^{6/7}} is equal to :

(where C ia a constant of integration) 

Option: 1

-\left ( \frac{x-3}{x+4} \right )^{-1/7}+C
 


Option: 2

\frac{1}{2}\left ( \frac{x-3}{x+4} \right )^{3/7}+C


Option: 3

\left ( \frac{x-3}{x+4} \right )^{1/7}+C

 


Option: 4

-\frac{1}{13}\left ( \frac{x-3}{x+4} \right )^{-13/7}+C


Answers (1)

best_answer

 

 

Integration Using Substitution -

The method of substitution is one of the basic methods for calculating indefinite integrals. 

Substitution - change of variable

\\\mathrm{To\;solve\;the\;integrate\;of\;the\;form}\\\\\mathrm{I=\int f\left ( g(x) \right )\cdot g'(x)\;dx,\;}\\\\\mathrm{\;where\;g(x)\;is\;continuously\;differentiable\;function.}\\\mathrm{put\;\;g(x)=t,\;\;g'(x)\;dx=dt}\\\mathrm{After\;substitution,\;we\;get\;\;\int f(t)\;dt.}\\\text{Evalute this integration and substitute back the value of }t.

 

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\\\int\left(\frac{x-3}{x+4}\right)^{\frac{-6}{7}} \frac{1}{(x+4)^{2}} d x\\ \\\text { Let } \frac{x-3}{x+4}=t^{7}\\ \\\frac{7}{(x+4)^{2}} d x=7 t^{6} d t\\ \\\int t^{-6} t^{6} d t=t+c\\\left ( \frac{x-3}{x+4} \right )^{\frac{1}{7}} +c

Correct Option (3)

Posted by

Deependra Verma

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