Advertisements
Advertisements
Question
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Advertisements
Solution
Let I = `int (1 + log x)^2/x*dx`
Put 1 + log x = t
∴ `(1)/x*dx` = dt
∴ I = `int t^3*dt = (1)/(4)t^4 + c`
= `(1)/(4)(1 + logx)^4 + c`.
APPEARS IN
RELATED QUESTIONS
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
Integrate the function in x log 2x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in x (log x)2.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
`int 1/(4x + 5x^(-11)) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int"e"^(4x - 3) "d"x` = ______ + c
Evaluate `int 1/(x(x - 1)) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
Find `int_0^1 x(tan^-1x) "d"x`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
`int(logx)^2dx` equals ______.
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(1-x)^-2 dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
`inte^(xloga).e^x dx` is ______
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following:
`intx^3e^(x^2)dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:
