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int(tanx)/(sqrt(sin^(4)x+cos^(4)x))dx is...

`int(tanx)/(sqrt(sin^(4)x+cos^(4)x))dx` is equal to

A

`ln(tan^(2)x)+sqrt(1+tan^(2)x)+C`

B

`secx+C`

C

`sqrt(1-tan^(2)x)+C`

D

None of the above

Text Solution

Verified by Experts

The correct Answer is:
D
`I=int(tanx)/(sqrt(sin^(4)x+cos^(4)x))dx=int(tanx.sec^(2)x)/(sqrt(1+tan^(4)x))dx`
`"Put "tanx=t`
`rArr" "sec^(2)x dx=dt`
`int(tanx)/(sqrt(sin^(4)x+cos^(4)x))dx=int(t)/(sqrt(1+t^(4)))dt`
`"Let "t^(2)=t rArr 2t dt = dy`
`therefore" "I=int(dy)/(2sqrt(1+y^(2)))=(1)/(2)ln|y+sqrt(y^(2)+1)|+C`
`=(1)/(2)ln|tan^(2)x+sqrt(tan^(4)x+1)|+C`
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