Advertisements
Advertisements
Question
Integrate the function in (x2 + 1) log x.
Advertisements
Solution
Let `I = int (x^2 + 1) log x dx`
`= int log x. (x^2 + 1) dx`
Integrating piecewise by taking (log x) as the first function, we get
`I = (log x) int (x^2 + 1) dx - int [d/dx log x int (x^2 + 1) dx] dx`
`= log x. (x^3/3 + x) - int 1/x . (x^3/3 + 1) dx`
`= (x^3/3 + x) log x - int (x^3/3 + 1) dx`
`= (x^3/3 + x) log x - (x^3/9 + x) + C`
`= (x^3/3 + x) log x - x^3/9 - x + C`
APPEARS IN
RELATED QUESTIONS
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Find :
`∫(log x)^2 dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Integrate the following w.r.t.x : log (log x)+(log x)–2
Evaluate the following.
`int x^3 e^(x^2)`dx
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int logx/(1 + logx)^2 "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
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 ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
Solution of the equation `xdy/dx=y log y` is ______
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate:
`intcos^-1(sqrt(x))dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
Which expression is the integration-by-parts form for a product \(f(x)g(x)\)?
Evaluate \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx.\]
