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int e^(tan^-1x)(1+x+x^2) d(cot^-1x) is e...

`int e^(tan^-1x)(1+x+x^2) d(cot^-1x)` is equal to

A

` - e^(tan^(-1)x)+c`

B

` e^(tan^(-1)x)+c`

C

`-x e^(tan^(-1)x)+c`

D

`x e^(tan^(-1)x)+c`

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int e^(tan^(-1)x) (1+x+x^(2))(-((1)/(1+x^(2)))dx)`
`= -int e^(tan^(-1)x)(1+(x)/(1+x^(2)))dx`
`= -int e^(tan^(-1)x)dx-int x(e^(tan^(-1)x))/(1+x^(2))dx`
`= -int e^(tan^(-1)x)dx-x e^(tan^(-1)x)+int e^(tan^(-1)x) dx+C`
`= -x e^(tan^(-1)x)+C`
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CENGAGE-INDEFINITE INTEGRATION-Exercise (Single)
  1. 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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  2. If I=inte^(-x)log(e^x+1)dx ,t h e nIe q u a l a+(e^(-x)+1)log(e^x+1)...

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  3. "If " int x e^(x) cosx dx=ae^(x)(b(1-x)sinx+cx cosx)+d, then

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  4. int x sinx sec^(3)x dx is equal to

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  5. int e^(tan^-1x)(1+x+x^2) d(cot^-1x) is equal to

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  6. int e^(x)((2 tanx)/(1+tanx)+cot^(2)(x+(pi)/(4)))dx is equal to

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  7. inte^x^4(x+x^3+2x^5)e^x^2dxis equal to 1/2x e^x^2e^x^4+c (b) 1/2x^2e...

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  8. The value of integral inte^x(1/(sqrt(1+x^2))+(1-2x^2)/(sqrt((1+x^2)^5)...

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  9. int e^(x)((x^(2)+1))/((x+1)^(2))dx is equal to

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  10. int ((x+2)/(x+4))^2 e^x dx is equal to

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  11. inte^(tanx)(sinx-secx)dx is equal to

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  12. int(cosec^2x-2005)/cos^[2005]x.dx

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  13. int(1+x-(1)/(x))e^(x+(1)/(x)) dx is equal to

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  14. int e^(sin^(-1)x)((log(e)x)/(sqrt(1-x^(2)))+(1)/(x))dx is equal to

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  15. Ifxf(x)=3f^2(x)+2,t h e nint(2x^2-12 xf(x)+f(x))/((6f(x)-x)(x^2-f(x))^...

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  16. The value of int((ax^2-b)dx)/(xsqrt(c^2x^2-(ax^2+b)^2)) is equal to

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  17. The value of int (dx)/((1+sqrtx)(sqrt(x-x^2))) is equal to

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  18. int(2sinx)/(3+sin2x)\ dx

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  19. 4 int (sqrt(a^(6)+x^(8)))/(x) dx is equal to

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  20. IfI(m , n)=intcos^m xsinn xdx ,t h e n7I(4,3)-4I(3,2)i se q u a lto c...

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