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int(1-7cos^(2)x)/(sin^(7)xcos^(2)x)dx=(f...

`int(1-7cos^(2)x)/(sin^(7)xcos^(2)x)dx=(f(x))/((sinx)^(7))+C`, then f(x) is equal to

A

`sinx`

B

`cos x`

C

`tanx`

D

`cotx`

Text Solution

Verified by Experts

The correct Answer is:
C
`int(1-7cos^(2)x)/(sin^(7)xcos^(2)x)dx=int((sec^(2)x)/(sin^(7)x)-(7)/(sin^(7)x))dx`
`=int(sec^(2)x)/(sin^(7)x)dx-int(7)/(sin^(7))=I_(1)+I_(2)`
`I_(2)=int(sec^(2)x)/(sin^(7)x)dx`
`=(tanx)/(sin^(7)x)+7int(tanxcosx)/(sin^(8)x)dx`
`=(tanx)/(sin^(7)x)-I_(2)`
`therefore" "I_(1)+I_(2)=(tanx)/(sin^(7)x)+C`
`rArr" "f(x)=tanx`
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