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
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`
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
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x sin x.
Integrate the function in x cos-1 x.
Integrate the function in e2x sin x.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following: `int logx/x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
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 (x- sinx)/(1 - cosx)*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
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)/(xsin^2(logx)`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Evaluate the following.
`int x^2 *e^(3x)`dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
`int 1/sqrt(x^2 - a^2)dx` = ______.
Solve: `int sqrt(4x^2 + 5)dx`
`int(logx)^2dx` equals ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(1-x)^-2 dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
`int1/(x+sqrt(x)) dx` = ______
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x 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 x^2 cos x dx`
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
