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  • The integral <img alt="\int \frac{\sin ^{2}x\cos ^{2}x}{\left ( \sin ^{3}x+\cos ^{3} x\right )^{2}}dx" src="https://learn.careers360.com/latex-image/?%5Cdpi%7B100%7D%20%5Cint%20%5Cfrac%7B%5Csin%20%

The integral \int \frac{\sin ^{2}x\cos ^{2}x}{\left ( \sin ^{3}x+\cos ^{3} x\right )^{2}}dx is equal to:

Option: 1

\frac{1}{\left ( 1+\cot ^{3} x\right )}+c


Option: 2

-\frac{1}{3\left ( 1+\\tan ^{3} x\right )}+c


Option: 3

\frac{\sin ^{3}x}{\left ( 1+\\cos ^{3} x\right )}+c


Option: 4

-\frac{\cos ^{3}x}{3\left ( 1+\\sin ^{3} x\right )}+c


Answers (1)

best_answer

As learnt in concept

Integration by substitution -

The functions when on substitution of the variable of integration to some quantity gives any one of standard formulas.

 

 

- wherein

Since \int f(x)dx=\int f(t)dt=\int f(\theta )d\theta all variables must be converted into single variable ,\left ( t\, or\ \theta \right )

 

 

  \int \frac{\sin^{2} x\cos^{2} x}{(\sin^{3} x+\cos^{3} x)^{2} }dx

\int \frac{\frac{\sin ^{2}x}{\cos ^{4}x}}{(1+\tan ^{3}x)^{2}}=\int \frac{\tan ^{2}x\sec ^{2}xdx}{(1+\tan ^{3}x)^{2}}

Put y=\tan x

\frac{1}{3}\int \frac{3y^{2}dy}{(1+y^{3})^{2}}= \frac{-1}{3}\times \frac{1}{(1+y^{3})}+C

\frac{-1}{3}\times \frac{1}{(1+\tan^{3} x)}+C

Posted by

Pankaj Sanodiya

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