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int(sin^3x)/((cos^4 x + 3 cos^2 x+1) tan...

`int(sin^3x)/((cos^4 x + 3 cos^2 x+1) tan^(-1) (sec x+ cos x))dx=`

A

`tan^(-1)(sec x+ cos x ) + C`

B

`log_e |tan^(-1) (sec x + cos x)| +C`

C

`1/((sec x + cos x)^2)+C`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
Let
`I=int(sin^3x)/((cos^4 x + 3 cos^2 x+1) tan^(-1) (sec x+ cos x))dx`
`rArr I=int((sin^3x)/(cos^2x))/((cos^2 x + 3+ sec^2 x) tan^(-1) (sec x + cosx ))dx`
`rArr I=int1/(1+(sec x + cos x )^2)xx(sin x(1-cos^2x))/(cos^2 x)xx1/(tan^(-1)(sec x + cos x ))dx`
`rArr I=int1/(tan^(-1) (sec x + cos x ))xx1/(1+(sec x + cos x )^2) xx (tan x sec x -sinx ) dx`
`rArr I=int1/(tan^(-1)(sec x + cos x )) d {tan^(-1) (sec x + cos x)}`
`rArr I=log|tan^(-1) (sec x + cos x ) |+C`
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