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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-INDEFINITE INTEGRATION-Exercise (Single)
  1. intx((lna^(x/2)/(3a^((5x)/2)b^(3x))+(lnb^b^x)/(2a^(2x)b^(4x)))dx(w h e...

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  2. int(3+2cosx)/((2+3cosx)^2)dxi se q u a lto ((sinx)/(3cosx+2))+c (b) ...

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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^(3)xcos^(5)x))=a sqrt(cot x)+bsqrt(tan^(3)x)+c, t...

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  6. "I f"int(dx)/(cos^3xsqrt(sin2x))=a(tan^2x+b)sqrt(tanx)+c

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  7. If int(dx)/((x+2)(x^(2)+1)) = alog|1+x^(2)|+btan^(-1)x+ 1/5log|x+2|+C,...

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  8. Ifint(3e^x-5e^(-1))/(4e^x+5e^(-x))dx=a x+bln(4e^x+5e^(-x))+C ,t h e n

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  9. If int f(x) sinx cosx dx=(1)/(2(b^(2)-a^(2))) "In " f(x)+c," then " f...

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  10. int(x^9)/((4x^2+1)^6) 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= sqrt(x^2+1/(x^2)+1)+C (sqrt(x^4...

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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 varp...

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