"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

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

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

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

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

Text Solution

Verified by Experts

The correct Answer is:

`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`

Topper's Solved these Questions

INDEFINITE INTEGRATION
CENGAGE ENGLISH|Exercise Exercises (Multiple Correct Answers Type)|17 Videos
INDEFINITE INTEGRATION
CENGAGE ENGLISH|Exercise Exercises (Linked Comprehension Type)|17 Videos
INDEFINITE INTEGRATION
CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 7.9|15 Videos
HYPERBOLA
CENGAGE ENGLISH|Exercise COMPREHENSION TYPE|2 Videos
INEQUALITIES AND MODULUS
CENGAGE ENGLISH|Exercise Single correct Answer|21 Videos

Similar Questions

Explore conceptually related problems

(d)/(dx)(int_(f(x))^(g(x)) phi(t)dt) is equal to

If d/dx (ϕ(x))=f(x) , then int_(1)^(2) f(x) dx is ______

Let f be a differentiable function satisfying f(xy)=f(x).f(y).AA x gt 0, y gt 0 and f(1+x)=1+x{1+g(x)} , where lim_(x to 0)g(x)=0 then int (f(x))/(f'(x))dx is equal to

Let f (x) be a twice differentiable function defined on (-oo,oo) such that f (x) =f (2-x)and f '((1)/(2 )) =f' ((1)/(4))=0. Then int _(-1) ^(1) f'(1+ x ) x ^(2) e ^(x ^(2))dx is equal to :

If (d)/(dx)f(x)=g(x) for a le x le b then, overset(b)underset(a)int f(x) g(x) dx equals

If f(x)=(x^2)/(|x|) , write d/(dx)(f(x))

Statement 1: If differentiable function f(x) satisfies the relation f(x)+f(x-2)=0AAx in R , and if (d/(dx)f(x))_(x=a)=b ,t h e n(d/(dx)f(x))_(x=a+4000)=b . Statement 2: f(x) is a periodic function with period 4.

Let f(x) is differentiable function satisfying 2int_(1)^(2)f(tx) dt = x+2, AA x in R Then int_(1)^(2)(8f(8x)-f(x)-21x) dx equal to

The integral inte^x(f(x)+f\'(x))dx can be solved by using integration by parts such that: I=inte^xf(x)dx+inte^xf\'(x)dx=e^xf(x)-inte^xf\'(x)dx+inte^xf\'(x)dx=e^xf(x)+C , and inte^(ax)(f(x)+(f\'(x))/a)dx=e^(ax)f(x)/a+C ,Now answer the question: inte^x x^x(2+logx)= (A) e^x x^xlogx+C (B) e^x+x^x+C (C) e^x x(logx)^2+C (D) e^x.x^x+C

According to Leibritz differentiation under the sign of integration can be performed as as below. (i) (d)/(dx)[int_(phi(x))^(Psi(x))f(t)dt]=f{Psi(x)}xx(d)/(dx){Psi(x)}-f{phi(x)}xx(d)/(dx){phi(x)} (ii) (d)/(dx)[int_(phi(x))^(Psi(x))f(x,t)dt]=int_(phi(x))^(Psi(x))(del)/(delx)(f(x,t)dt)+f(x,Psi(x))xx(d)/(dx)Psi(x)-f(x,phi(x))xx(d)/(dx)(phi(x)) int_(x^(2))^(x^(3))cost^(2)dt has the derivative

CENGAGE ENGLISH-INDEFINITE INTEGRATION-EXERCISES (Single Correct Answer Type)

intx((lna^(x/2)/(3a^((5x)/2)b^(3x))+(lnb^b^x)/(2a^(2x)b^(4x)))dx(w h e...
Text Solution
|
int(3+2cosx)/((2+3cosx)^2)dx is equal to (a) ((sinx)/(3cosx+2))+c (b...
Text Solution
|
"If " (d)/(dx)f(x)=f'(x), " then " int(xf'(x)-2f(x))/(sqrt(x^(4)f(x)))...
Text Solution
|
The value of the integral int((1-costheta)^(2/7))/((1+costheta)^(9/7))...
Text Solution
|
If int(dx)/(sqrt(sin^(3)xcos^(5)x))=a sqrt(cot x)+bsqrt(tan^(3)x)+c, t...
Text Solution
|
"I f"int(dx)/(cos^3xsqrt(sin2x))=a(tan^2x+b)sqrt(tanx)+c
Text Solution
|
If int(dx)/((x+2)(x^(2)+1)) = alog|1+x^(2)|+btan^(-1)x+ 1/5log|x+2|+C,...
Text Solution
|
Ifint(3e^x-5e^(-x))/(4e^x+5e^(-x))dx=a x+bln(4e^x+5e^(-x))+C ,t h e n ...
Text Solution
|
If intf(x)sinxcosxdx=1/(2(b^2-a^2))lnf(x)+c ,then f(x) is equal to
Text Solution
|
int(x^(9))/((4x^(2)+ 1)^(6))dx is equal to
Text Solution
|
If int1/(xsqrt(1-x^3))dx=alog|(sqrt(1-x^3)-1)/(sqrt(1-x^3)+1)|+b , th...
Text Solution
|
The value of the integral int(x^2+x)(x^(-8)+2x^(-9))^(1/(10))dx is 5/(...
Text Solution
|
int(x^3dx)/(sqrt(1+x^2))i se q u a lto 1/3sqrt(1+x^2)(2+x^2)+C 1/3...
Text Solution
|
If I=int (dx)/((a^(2)-b^(2)x^(2))^(3//2)), then I equals
Text Solution
|
int((x^4-1)dx)/(x^2sqrt(x^4+x^2+1))
Text Solution
|
int(sqrt(x^2+1))/(x^4)dx=
Text Solution
|
int (1+2x^(6))/((1-x^(6))^(3//2))dx is equal to
Text Solution
|
The integral int[2x^[12]+5x^9]/[x^5+x^3+1]^3.dx is equal to-
Text Solution
|
If In=int( lnx)^n dx then In+nI(n-1)
Text Solution
|
int e^x {f(x)-f'(x)}dx= phi(x), then int e^x f(x) dx is
Text Solution
|