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Prove that, int(cos 5x + cos4x)/(1-2cos ...

Prove that, `int(cos 5x + cos4x)/(1-2cos 3x)dx= -(1/2 sin2x + sinx)+c`

A

`(sin2x)/(2)+cosx+c`

B

`(sin2x)/(2)-cosx+c`

C

`-(sin2x)/(2)-sinx+c`

D

`(sin2x)/(2)-cosx+c`

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int(cos5x+cos4x)/(1-2cos3x)dx=int(2"cos"(9x)/(2)"cos"(x)/(2))/(1-2(2"cos"^(2)(3x)/(2)-1))dx`
`=int(2"cos"(9x)/(2)"cos"(x)/(2))/(3-4"cos"^(2)(3x)/(2))dx`
`=int(2"cos"(9x)/(2)"cos"(x)/(2)"cos"(3x)/(2))/(3"cos"(3x)/(2)-4"cos"^(3)(3x)/(2))dx`
`=int(2"cos"(9x)/(2)"cos"(3x)/(2)"cos"(x)/(2))/(-"cos"(9x)/(2))dx`
`=-int 2"cos"(3x)/(2)"cos"(x)/(2)dx`
`=-int(cos2x+cosx)dx`
`=-(sin2x)/(2)-sinx+c`
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CENGAGE PUBLICATION-INDEFINITE INTEGRATION-EXERCISES (Single Correct Answer Type)
  1. IfI=int(dx)/(secx+cos e cx),t h e nIe q u a l s

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  2. int(sinx)/(sin(x-(pi)/(4)))dx " is equal to "

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  3. Prove that, int(cos 5x + cos4x)/(1-2cos 3x)dx= -(1/2 sin2x + sinx)+c

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  4. IfI=intsqrt((5-x)/(2+x))dx ,t h e nIe q u a l sqrt(x+2)sqrt(5+x)+3si...

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  5. int(sin2x)/(sin5xsin3x)dx is equal to

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  6. int(dx)/(x(x^n+1)) is equal to

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  7. Evaluate:int (dx)/(sqrt (sin^3 xsin(x+alpha)

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  8. int (px^(p+2q-1)-qx^(q-1))/(x^(2p+2q)+2x^(p+q)+1) dx is equal to (1...

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  9. If y=int1/(1+x^2)^(3/2)dx and y=0 when x=0 , then value of y when x=1 ...

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  10. int sqrt(x)(1+x^(1//3))^(4)dx " is equal to "

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  11. int(In(tanx))/(sinx cosx)dx " is equal to "

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  12. If m is a non-zero number and int (x^(5m-1)+2x^(4m-1))/(x^(2m)+x^m+1)^...

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  13. If l^r(x) means logloglog...... x , the log being repeated r times, th...

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  14. Evaluate: int (sqrt(tanx)+sqrtcotx)dx

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  15. IfI=int(sin2x)/((3+4cosx)^3)dx ,t h e nIe q u a l s (3cosx+8)/((3+4co...

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  16. int(ln((x-1)/(x+1)))/(x^2-1)dxi se q u a lto 1/2(ln((x-1)/(x+1)))^2+...

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  17. intsqrt(e^x-1)dx is equal to (a) 2[sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)]+...

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  18. Evaluate: int((1+x^2)/(x^2-x^4))dx

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  19. int(sqrt(x^2+10 x+24))/(x+5)dx is equal to

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  20. The value of int(1+logx)/(sqrt((x^(x))^(2)-1))dx " is "

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