The integral `int (sin^(2)x cos^(2)x)/((sin^(5)x+cos^(3)x sin^(2)x+sin^(3)x cos^(2)x+cos^(5)x)^(2))dx` is equal to (where C is a constant of integration)

Options

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`

Answer

C

Explanation

`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`