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If int (sqrtx)^5/((sqrtx)^7+x^6) dx= alo...

If `int (sqrtx)^5/((sqrtx)^7+x^6) dx= alog(x^k/(1+x^k))+c` then `a` and `k` are

A

`k=-2, f(x)=cot^(-1)x, g(x)=sqrt("cosec"x-1)`

B

`k=-2, f(x)=tan^(-1)x, g(x)=sqrt("cosec"x-1)`

C

`k=2, f(x)=tan^(-1)x, g(x)=(cotx)/(sqrt("cosec"x-1))`

D

`k=2, f(x)=cot^(-1)x, g(x)=(cotx)/(sqrt("cosec"x+1))`

Text Solution

Verified by Experts

The correct Answer is:
B, D
`I= int sqrt("cosec"x+1)dx=int(cotx)/(sqrt("cosec"x-1))dx`
` "Put cosec"x-1=t^(2) or -"cosec"x cotx dx =2t dt`
` :. I= -int(-cotx"cosec"x)/("cosec"x sqrt("cosec"x -1))dx`
`=-int (2dt)/(1+t^(2))= -2tan^(-1)t+c`
`= -2tan^(-1)sqrt("cosec"x-1)+C`
`= -2[(pi)/(2)-cot^(-1)sqrt("cosec"x-1)]+C`
`=2 cot^(-1)sqrt("cosec"x-1)+C`
`=2"cot"^(-1)(cotx)/(sqrt("cosec"x+1))+C`
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