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if int(1-5sin^2x)/(cos^5xsin^2x)dx=f(x)/...

if `int(1-5sin^2x)/(cos^5xsin^2x)dx=f(x)/(cos^5x)+c` then `f(x)`

A

`-cotx`

B

`-"cosec "x`

C

`"cosec "x`

D

cot x

Text Solution

Verified by Experts

The correct Answer is:
d
Let `I=int(1-5 sin^(2)x)/(cos^(5)x sin^(2)x)dx` . Then,
`I=int((cos^(2)x+sin^(2)x)-5sin^(2)x)/(cos^(5)x sin ^(2)x)dx`
`rArr I=int(cos^(2)x-4sin^(2)x)/(cos^(5)xsin^(2)x)dx=int(cos^(5)x-4sin^(2)xcos^(3)x)/(cos^(8) xsin^(2)x)dx`
`rArrI=int(d)/(dx)((1)/(cos^(4)xsinx))=(1)/(cos^(4)xsinx)+C`
`rArrI=(cotx)/(cos^(5)x)+C`
`rArr(f(x))/(cos^(5)x)+C=(cotx)/(cos^(5)x)+C`
`rArr f(x) = cotx`.
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