Advertisements
Advertisements
प्रश्न
Integrate the function in `(xe^x)/(1+x)^2`.
Advertisements
उत्तर
Let `I = int (xe^x)/((1 + x)^2) dx`
`= int ((x + 1 - 1) e^x)/((1 + x)^2) dx`
`= int 1/((1 + x)) . e^x dx - (e^x - 1)/((1 + x)^2) dx`
`= 1/((1 + x)). e^x - int (-1)/((1 + x^2)).e^x dx - int e^x/((1 + x^2)) dx + C`
`= e^x/(1 + x) + int e^x/((1 + x)^2) dx - int e^x/((1 + x)^2) dx + C`
`= e^x/(1 + x) + C`
APPEARS IN
संबंधित प्रश्न
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Find :
`∫(log x)^2 dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : e2x sin x cos x
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
`int (sinx)/(1 + sin x) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Solve: `int sqrt(4x^2 + 5)dx`
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.
Solution: (x2 + y2) dx - 2xy dy = 0
∴ `dy/dx=(x^2+y^2)/(2xy)` ...(1)
Puty = vx
∴ `dy/dx=square`
∴ equation (1) becomes
`x(dv)/dx = square`
∴ `square dv = dx/x`
On integrating, we get
`int(2v)/(1-v^2) dv =intdx/x`
∴ `-log|1-v^2|=log|x|+c_1`
∴ `log|x| + log|1-v^2|=logc ...["where" - c_1 = log c]`
∴ x(1 - v2) = c
By putting the value of v, the general solution of the D.E. is `square`= cx
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`int e^(logcosx)dx`
Evaluate the following.
`intx^3e^(x^2) dx`
Evaluate:
`int x^2 cos x dx`
