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"If " (d)/(dx)f(x)=f'(x), " then " int(x...

`"If " (d)/(dx)f(x)=f'(x), " then " int(xf'(x)-2f(x))/(sqrt(x^(4)f(x)))dx ` is equal to

A

`(x^(2))/(f(x))+c`

B

`|x|f'(x)+c`

C

`(2sqrt(f(x)))/(|x|)+c`

D

`|x|sqrt(f(x))+c`

Text Solution

Verified by Experts

The correct Answer is:
C
`I=int(xf'(x)-2f(x))/(x^(2)sqrt(f(x)))dx `
`=int((x^(2)f'(x)-2xf(x))/(x^(4)))/(sqrt((f(x))/(x^(2))))dx`
`"Put " (f(x))/(x^(2))=t^(2) `
` :. (x^(2)f'(x)-2xf(x))/(x^(4))dx=2t dt `
` :. I=int(2t dt)/(t)=2t+c=(2sqrt(f(x)))/(|x|)+c`
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CENGAGE PUBLICATION-INDEFINITE INTEGRATION-EXERCISES (Single Correct Answer Type)
  1. intx(lna^(a^(x/2))/(3a^((5x)/2)b^(3x))+(lnb^(b^x))/(2a^(2x)b^(4x)))dx(...

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  2. int(3+2cosx)/((2+3cosx)^2)dx is equal to (a) ((sinx)/(3cosx+2))+c ...

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

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

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  5. If int(dx)/(sqrt(sin^3xcos^5x))=asqrt(cotx)+bsqrt(tan^3)x+c , then a=-...

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  6. Ifint(dx)/(cos^3xsqrt(sin2x))=a(tan^2x+b)sqrt(tanx)+c ,t h e n a=(sq...

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  7. Evaluate: int dx/( (x + 2)(x^2 + 1) )

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  8. Ifint(3e^x-5e^(-x))/(4e^x+5e^(-x))dx=a x+bln(4e^x+5e^(-x))+C ,t h e n ...

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  9. If intf(x)sinxcosxdx=1/(2(b^2-a^2))lnf(x)+c ,then f(x) is equal to

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  10. int(x^(5))/((4x^(2)+ 1)^(4))dx is equal to

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

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

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  15. int(x^4-1)/(x^2sqrt(x^4+x^2+1))dx= (a) sqrt(x^2+1/(x^2)+1)+C (b) (...

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  16. int(sqrt(x^2+1))/(x^4)dx=

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  17. int (1+2x^(6))/((1-x^(6))^(3//2))dx is equal to

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  18. The integral int[2x^[12]+5x^9]/[x^5+x^3+1]^3.dx is equal to-

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  19. If In=int( lnx)^n dx then In+nI(n-1)

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  20. Let inte^x{f(x)-f^(prime)(x)}dx=varphi(x)dot Then inte^xf(x)dx is

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