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int(cos2x)/((e^(-x)+cosx)sqrt(1+sin2x))d...

`int(cos2x)/((e^(-x)+cosx)sqrt(1+sin2x))dx,x in(0,(pi)/(2)) ` is equal to

A

`log_(e)|1+e^(x)sinx|+c`

B

`log_(e)|e^(x)+cosx|+c`

C

`log_(e)|1+e^(x)cosx|-x+c`

D

`log_(e)|1+e^(x)cosx|+c`

Text Solution

Verified by Experts

The correct Answer is:
D
`int(cos2x)/((e^(-x)+cosx)sqrt(1+sin2x))dx`
`=int(cos^(2)x-sin^(2)x)/((e^(-x)+cosx)sqrt((cosx+sinx)^(2)))dx`
`=int(cosx-sinx)/((e^(-x)+cosx))dx`
`=int(e^(x)(cosx-sinx))/(1+e^(x)cosx)dx`
`=int((1+e^(x)cosx)')/(1+e^(x)cosx)dx`
`=log_(e)|1+e^(x)cosx|+c`
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CENGAGE-INDEFINITE INTEGRATION-Exercise (Single)
  1. int(x+2)/((x^(2)+3x+3)sqrt(x+1))dx " is equal to"

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  2. int(sec xdx)/(sqrt(sin(2x+A)+sinA)) " is equal to "

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  3. int(cos2x)/((e^(-x)+cosx)sqrt(1+sin2x))dx,x in(0,(pi)/(2)) is equal t...

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  4. int(cos4x-1)/(cotx-tanx)dx is equal to

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  5. If int(dx)/(x^2(x^n+1)^((n-1)/n))=-(f(x))^(1/n)+C then f(x) is (...

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  6. int sqrt((cosx-cos^3x)/(1-cos^3x))dx is equal to

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  7. intx((lna^(x/2)/(3a^((5x)/2)b^(3x))+(lnb^b^x)/(2a^(2x)b^(4x)))dx(w h e...

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  8. int(3+2cosx)/((2+3cosx)^(2))dx is equal to

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  9. "If " (d)/(dx)f(x)=f'(x), " then " int(xf'(x)-2f(x))/(sqrt(x^(4)f(x)))...

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  10. The value of the integral int((1-costheta)^(2/7))/((1+costheta)^(9/7))...

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  11. If int(dx)/(sqrt(sin^(3)xcos^(5)x))=a sqrt(cot x)+bsqrt(tan^(3)x)+c, t...

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

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

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

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  16. int(x^9)/((4x^2+1)^6) dx is equal to

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

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

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