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int(xcosx+1)/(sqrt(2x^(3)e^(sinx)+x^(2))...

`int(xcosx+1)/(sqrt(2x^(3)e^(sinx)+x^(2)))dx`

A

`log|(sqrt(2xe^(sinx)+1)-1)/(sqrt(2xe^(sinx)+1)+1)|+C`

B

`log|(sqrt(2xe^(sinx)-1)+1)/(sqrt(2xe^(sinx)+1)+1)|+C`

C

`log|(sqrt(2xe^(sinx)+1)+1)/(sqrt(2xe^(sinx)-1)+1)|+C`

D

`log|(sqrt(2xe^(sinx)+1)+1)/(sqrt(2xe^(sinx)-1)-1)|+C`

Text Solution

Verified by Experts

The correct Answer is:
A
`l=int((x cos x+1)e^(sinx)dx)/(e^(sinx)x sqrt(2xe^(sinx)+1))`
`"Put "2xe^(sinx)+1=t^(2)`
`rArr" "2[e^(sinx)+e^(sinx)cosx.x]dx=2tdt`
`rArr" "e^(sinx)(1=x cos x)dx= tdt`
`therefore" "l=int(tdt)/(((t^(2)-1))/(2)xt)`
`l=2int(dt)/(t^(2)-1)=log|(t-1)/(t+1)|+C`
where, `t^(2)=2xe^(sinx)+1`
`l=log|(sqrt(2xe^(sinx)+1)-1)/(sqrt(2xe^(sinx)+1)-1)|+C`
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