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

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  4. If int(dx)/((x+2)(x^(2)+1)) = alog|1+x^(2)|+btan^(-1)x+ 1/5log|x+2|+C,...

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  5. Ifint(3e^x-5e^(-1))/(4e^x+5e^(-x))dx=a x+bln(4e^x+5e^(-x))+C ,t h e n ...

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  6. If int f(x) sinx cosx dx=(1)/(2(b^(2)-a^(2))) "In " f(x)+c," then " f...

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  8. If int1/(xsqrt(1-x^3))dx=alog|(sqrt(1-x^3)-1)/(sqrt(1-x^3)+1)|+b ,t h...

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  9. The value of the integral int(x^2+x)(x^(-8)+2x^(-9))^(1/(10))dx is 5/(...

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  10. int(x^3dx)/(sqrt(1+x^2))i se q u a lto 1/3sqrt(1+x^2)(2+x^2)+C 1/3...

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  11. If I=int (dx)/((a^(2)-b^(2)x^(2))^(3//2)), then I equals

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  12. int((x^4-1)dx)/(x^2sqrt(x^4+x^2+1))

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  13. int(sqrt(x^2+1))/(x^4)dx=

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  15. The integral int(2x^(12)+5x^9)/((x^5+x^3+1)^3)dx is equal to:

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  16. If In=int( lnx)^n dx then In+nI(n-1)

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  18. For x >1,int sin^(- 1)((2x)/(1+x^2))dx is equal to

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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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  20. Ifintxlog(1+1/x)dx=f(x)log(x+1)+g(x)x^2+A x+C , then f(x)=1/2x^2 (b) ...

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