Advertisements
Advertisements
प्रश्न
Integrate the function in x log x.
Advertisements
उत्तर
Let `I = int x log x dx`
`= log x int x dx - int [d/dx (log x) int x dx] dx`
`= log x (x^2/2) - int (1/x * x^2/2) dx`
`= x^2/2 log x - 1/2 int x dx + C`
`= x^2/2 log x -1/2 xx x^2/2 + C`
`= x^2/2 log x - 1/4 x^2 + C`
APPEARS IN
संबंधित प्रश्न
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
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in x sin 3x.
Integrate the function in x (log x)2.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int x.cos^3x.dx`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
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
Integrate the following w.r.t.x : log (x2 + 1)
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
`int logx/(1 + logx)^2 "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int1/(x+sqrt(x)) dx` = ______
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx.\]
Which substitution can also be used before integrating by parts for \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx?\]
Evaluate \[\int e^x\sin x\,dx.\]
For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?
When applying integration by parts to \(\log x\) and inverse trig, what should they be multiplied by?
