Advertisements
Advertisements
Question
Integrate the function in x2 log x.
Advertisements
Solution
Let `I = int x^2 log x dx`
`= log (x) (x^3/3) - int [d/dx (log x) (x^3/3)] dx`
`= log x. x^3/3 - int 1/x. x^3/3 dx`
`= x^3/3 log x - 1/3 int x^2 dx`
`= x^3/3 log x - 1/3. x^3/3 + C`
`= x^3/3 log x - x^3/9 + C`
APPEARS IN
RELATED QUESTIONS
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Integrate the function in x sin−1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Choose the correct options from the given alternatives :
`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : log (x2 + 1)
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 (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int 1/(4x + 5x^(-11)) "d"x`
`int 1/x "d"x` = ______ + c
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
`int 1/sqrt(x^2 - a^2)dx` = ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
`intsqrt(1+x) dx` = ______
`int1/(x+sqrt(x)) dx` = ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
