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IfI=int(dx)/(secx+cos e cx),t h e nIe q ...

`IfI=int(dx)/(secx+cos e cx),t h e nIe q u a l s`

A

`(1)/(2)(cosx+sinx-(1)/(sqrt(2))log("cosec"x-cosx))+C`

B

`(1)/(2)(sinx-cosx-(1)/(sqrt(2))log|"cosec"x-cotx|)+C`

C

`(1)/(sqrt(2))(sinx+cosx+(1)/(2)log|"cosec"x-cosx|)+C`

D

`(1)/(2)[sinx-cosx]-(1)/(sqrt(2))log|"cosec"(x+pi//4)-cot(x+pi//4)|+C`

Text Solution

Verified by Experts

The correct Answer is:
D
`I=int(sinx cosx)/(sinx+cosx)dx`
`=(1)/(2)int((sinx+cosx)^(2)-1)/(sinx+cosx)dx`
`=(1)/(2)int[sinx+cosx-(1)/(sqrt(2)sin(x+pi//4))]dx`
`=(1)/(2)[sinx-cosx]-(1)/(2sqrt(2))log|"cosec"(x+pi//4)-cot(x+pi//4)|+C`
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CENGAGE PUBLICATION-INDEFINITE INTEGRATION-EXERCISES (Single Correct Answer Type)
  1. If intsqrt(1+sinx)f(x)dx=2/3(1+sinx)^(3/2)+c ,t h e nf(x)e q u a l co...

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  2. int(sqrt(x-1))/(xsqrt(x+1))dx is e q u a l to ln|x-sqrt(x^...

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  3. IfI=int(dx)/(secx+cos e cx),t h e nIe q u a l s

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

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

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  6. 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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  7. int(sin2x)/(sin5xsin3x)dx is equal to

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

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

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

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

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

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

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

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

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

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

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

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