Advertisements
Advertisements
प्रश्न
Integrate the function in x (log x)2.
Advertisements
उत्तर
Let `I = int x (log x)^2 dx`
`= int (log x)^2 * x dx`
`= (log x)^2 int x dx - int [d/dx (log x)^2 * int x dx] dx`
`= x^2/2 (log x)^2 - int (log x) * x dx + C`
`= x^2/2 (log x)^2 - [ (log x) * x^2/2 - int 1/x * x^2/2 dx]`
`= x^2/2 (log x)^2 - x^2/2 log x + 1/2 int x dx`
`= x^2/2 (log x)^2 - x^2/2 log x + 1/2 int*x^2/2 + C`
`= x^2 (log x)^2 - x^2/2 log x + 1/2 * x^2/2 + C`
APPEARS IN
संबंधित प्रश्न
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^2e^x`.
Integrate the function in x log x.
Integrate the function in (x2 + 1) log x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
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 : `int x^3.tan^-1x.dx`
Evaluate the following : `int e^(2x).cos 3x.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 : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
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 : sec4x cosec2x
Evaluate the following.
`int x^2 *e^(3x)`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(5 - 16"x"^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int sin4x cos3x "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int1/sqrt(x^2 - a^2) dx` = ______
`intsqrt(1+x) dx` = ______
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
`int logx dx = x(1+logx)+c`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
