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"I f"int(dx)/(cos^3xsqrt(sin2x))=a(tan^2...

`"I f"int(dx)/(cos^3xsqrt(sin2x))=a(tan^2x+b)sqrt(tanx)+c`

A

`a=(sqrt(2))/(5), b=(1)/(sqrt(5))`

B

`a=(sqrt(2))/(5), b=5`

C

`a=(sqrt(2))/(5), b= -(1)/(sqrt(5))`

D

`a=(sqrt(2))/(5), b= sqrt(5)`

Text Solution

Verified by Experts

The correct Answer is:
B
`I=int(dx)/(cos^(3)xsqrt(sin2x))`
`=int(dx)/(cos^(3)xsqrt((2sinx cosx)/(cos^(2)x)cos^(2)x))`
`=int(sec^(4)x)/(sqrt(2tanx))=(1)/(sqrt(2))int(sec^(2)x(1+tan^(2)x))/(sqrt(tanx))dx`
Let `t=sqrt(tanx)`
or `dt=(sec^(2)xdx)/(2sqrt(tanx))`
` :. dt =(2)/(sqrt(2))int(1+t^(4))dt=sqrt(2)(1+(t^(5))/(5))+C`
`=(sqrt(2))/(5) t(t^(4)+5)+C`
`=(sqrt(2))/(5)sqrt(tanx)(tan^(2)x+5)+C`
or `a=(sqrt(2))/(5), b=5`
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CENGAGE-INDEFINITE INTEGRATION-Exercise (Single)
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  2. If int(dx)/(sqrt(sin^(3)xcos^(5)x))=a sqrt(cot x)+bsqrt(tan^(3)x)+c, t...

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  19. If int x((ln(x+sqrt(1+x^2)))/sqrt(1+x^2)) dx=asqrt(1+x^2)ln(x+sqrt(1+x...

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