Advertisements
Advertisements
प्रश्न
Integrate the function in (x2 + 1) log x.
Advertisements
उत्तर
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
संबंधित प्रश्न
Integrate the function in x sin 3x.
Integrate the function in x2 log x.
Integrate the function in `(xe^x)/(1+x)^2`.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following : `int x^2tan^-1x.dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
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 "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int "dx"/(5 - 16"x"^2)`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int(x + 1/x)^3 dx` = ______.
`int 1/x "d"x` = ______ + c
`int 1/sqrt(x^2 - 8x - 20) "d"x`
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 ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int1/sqrt(x^2 - a^2) dx` = ______
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate:
`int e^(logcosx)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:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3 e^(x^2) 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).
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
