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

B

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

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int(sin^(2)xcos^(2)x)/((sin^(5)x+cos^(3)xsin^(2)x+sin^(3)xcos^(2)x+cos^(5)x)^(2))dx `
Dividing numerator and denominator by ` cos^(10)x,` we get
`I=int(tan^(2)xsec^(6)xdx)/((tan^(5)x+tan^(2)x+tan^(3)x+1)^(2))`
` =int(tan^(2)xsec^(6)x)/((1+tan^(2)x)^(2)(1+tan^(3)x)^(2))dx`
`=int(tan^(2)xsec^(6)x)/((sec^(2)x)^(2)(1+tan^(3)x)^(2))dx`
`=int(tan^(2)xsec^(2)x)/(1+tan^(3)x)^(2)dx`
`=(1)/(3)int(3tan^(2)xsec^(2)x)/((1+tan^(3)x)^(2))dx`
`=(1)/(3)int((1+tan^(3)x)')/((1+tan^(3)x)^(2))dx`
`=(-1)/(3(1+tan^(3)x))+C`
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