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int \ (sin^2x cos^2x)/(sin^5x+cos^3x sin...

`int \ (sin^2x cos^2x)/(sin^5x+cos^3x sin^2x + sin^3x cos^2x + cos^5x)^2 \ dx`

A

`(1)/(3(1+tan^(3)x))+C`

B

`(-1)/(3(1+tan^(3)x))+C`

C

`(1)/(1+cot^(3)x)+C`

D

`(-1)/(1+cot^(3)x)+C`

Text Solution

Verified by Experts

The correct Answer is:
B
We have, `I = int(sin^(2)x*cos^(2)x)/((sin^(6)x + cos^(3)x*sin^(2)x+sin^(3)x*cos^(2)x+cos^(5)x)^(2))dx`
` = int(sin^(2)x cos^(2)x)/{{sin^(3)x(sin^(2) x + cos^(2)x)+cos^(3)x(sin^(2)x +cos^(2)x)}^(2))dx`
`=int(sin^(2)x cos^(2) x)/((sin^(3)x + cos^(3)x)^(2))dx=int(sin^(2)x cos^(2)x)/(cos^(6)x(1+tan^(3)x)^(2))dx`
`=int(tan^(2)x sec^(2)x)/((1+tan^(3)x)^(2))dx`
Put `tan^(3) x = t rArr 3 tan^(2)x sec^(2) xdx = dt`
`therefore" "I = (1)/(3)int(dt)/((1+t)^(2))`
`rArr" "I = (-1)/(3(1+t))+C rArr I=(-1)/(3(1+tan^(3)x))+C`
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