Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int x^2 sin 3x dx`
Advertisements
उत्तर
Let I = `int x^2 sin 3x dx`
= `x^2 int sin 3x.dx - int [d/dx (x^2) int sin 3x.dx]dx ...[∵ int uv.dx = uintv.dx - int[(du)/(dx) int v.dx]dx]`
= `x^2(-(cos3x)/3) - int2x(-(cos3x)/3).dx`
= `-x^2/3 cos3x + (2)/(3) int x cos 3x dx`
= `-x^2/3 cos3x + (2)/(3)[x int cos 3x dx - int {d/dx (x) int cos 3x .dx} .dx] ...[∵ int uv.dx = uintv.dx - int[(du)/(dx) int v.dx]dx]`
= `-x^2/3 cos3x + 2/3[(xsin3x)/(3) - int 1. (sin3x)/(3).dx]`
= `-x^2/3 cos3x + (2 x sin 3x)/9 - (2)/(9) int (sin 3x)/3 dx`
= `-x^2/3 cos3x + (2 x sin 3x)/9 - (2)/(9) ((- cos3x)/3) + c`
= `-x^2/3 cos3x + (2 x sin 3x)/9 + (2 cos 3x)/27 + c`
संबंधित प्रश्न
Integrate the function in x sin 3x.
Integrate the function in x log 2x.
Integrate the function in x sin−1 x.
Integrate the function in x cos-1 x.
Integrate the function in (sin-1x)2.
Integrate the function in tan-1 x.
Integrate the function in ex (sinx + cosx).
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Evaluate the following:
`int sec^3x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
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 functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
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` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*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 with respect to the respective variable : cos 3x cos 2x cos x
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int 1/sqrt(2x^2 - 5) "d"x`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int(x + 1/x)^3 dx` = ______.
`int logx/(1 + logx)^2 "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
`int 1/sqrt(x^2 - 9) dx` = ______.
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
`inte^(xloga).e^x dx` is ______
Evaluate:
`int (logx)^2 dx`
Evaluate `int tan^-1x dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate:
`int1/(x^2 + 25)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 ______.
