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int (x^2(xsec^2x+tanx))/(xtanx+1)^2...

`int (x^2(xsec^2x+tanx))/(xtanx+1)^2`

A

`(-x)/((tanx+1))+2" In"|x sinx+cosx|+C`

B

`(-x^(2))/((xtanx+1))+2|sinx+cosx|+C`

C

`(-x^(2))/((xtanx+1))+2|sinx+cosx|+C`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
b
Let `I= intx^(2)(xsec^(2)x+tanx)/((xtanx+1)^(2))dx` . Then,
`I=int underset(I)(x^(2))(1)/(underset(II)((tan x+1))^(2))d(xtan+1)`
`rArrI=(-x^(2))/((xtanx+1))+2int(x)/((xtanx+1))dx`
`rArrI=(-x^(2))/((xtanx+1))+2int(xcosx)/((x sinx+cosx))dx`
`rArrI=(-x^(2))/((xtanx+1))+2 " In "|x sinx+cosx|+C`
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OBJECTIVE RD SHARMA ENGLISH-INDEFINITE INTEGRALS-Solved Example
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  2. inte^(x^(4))(x+x^(3)+2x^(5))e^(x^(2))dx is equal to

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  3. int(sin(101x).sin^(99)x)dx equals

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  4. Suppose int(1-7cos^2x)/(sin^7xcos^2x)dx=(g(x))/(sin^7x)+c where C is a...

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  6. If int 2/(2-x)^2 ((2-x)/(2+x))^(1//3)\ dx = lambda ((2+x)/(2-x))^mu + ...

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  7. Let f(x) be a quadratic function such that f(0)=1 and int(f(x))/(x^2(x...

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  8. int (x dx)/(sqrt(1+x^(2)+sqrt((1+x^(2))^(3))) is equal to

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

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  10. intx^(27)(1+x+x^(2))^(6)(6x^(2)+5x+4)dx=

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  11. Let f (x) be a quadratic function such that f (0) =1 and f(-1)=4, if...

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  12. Evaluate int(e^(tan^(-1)x))/((1+x^(2)))[(sec^(-1)sqrt(1+x^(2)))^(2)+"c...

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  13. int (x^2(xsec^2x+tanx))/(xtanx+1)^2

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  14. int (mx^(m+2n-1)-nx^(n-1))/(x^(2m+2n)+2x^(m+n)+1)dx is equal to

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  15. Evaluate int(dx)/(tanx+cotx+secx+cosecx).

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  16. int (x^(2)(1-"ln"x))/(("ln"^(4)x-x^(4))dx is equal to

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

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  18. If int(dx)/(cos^(3)xsqrt(sin2x))=a(tan^(2)x+b)sqrt(tanx)+C, then

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  19. "If"int(dx)/(x^(3)(1+x^(6))^(2/3))=xf(x)(1+x^(6))^(1/3)+C where, C is ...

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  20. int(dx)/((1+sqrt(x))sqrt((x-x^(2)))) is equal to

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