Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Advertisements
उत्तर
Let I = `int e^-x cos 2x.dx`
∫ uv dx = u ∫ v dx – ∫ (u' ∫ v dx) dx.
I = `cos 2x int e^-x dx – int int e^(-x). d/dx cos 2x. dx`
I = `cos 2x. (e^-x)/(d/dx (-x)) – int(e^-x)/(d/dx (- x)). (- sin 2x. d/dx 2x) dx`
I = `- cos 2x. e^-x – int (- e^(-x)) . (- 2sin 2x) dx`
I = `- cos 2x. e^-x – 2 int e^(-x). sin 2x dx`
I = `- cos 2x. e^-x - 2 [sin 2x. int e^-x dx - int int e^(-x) dx. d/dx sin 2x. dx]`
I = `- cos 2x. e^-x - 2 sin 2x. (e^-x)/(- 1) + 2 int (e^-x)/(- 1). cos 2x. 2. dx`
I = `- cos 2x. e^-x + 2 sin 2x. (e^-x) - 2 int 2. e^(-x). cos 2x.dx`
I = `- cos 2x. e^-x + 2 sin 2x. (e^-x) - 4 int e^(-x). cos 2x.dx`
I = `- cos 2x. e^-x + 2 sin 2x. (e^-x) - 4I`
I + 4I = `- cos 2x. e^-x + 2 sin 2x. (e^-x)`
5I = `e^-x (2. sin 2x - cos 2x)`
I = `e^-x/5 (2. sin 2x - cos 2x) + C`
Notes
Let I = `int e^-x cos 2x.dx`
∫ uv dx = u ∫ v dx – ∫ (u' ∫ v dx) dx.
I = `cos 2x int e^-x dx – int int e^(-x). d/dx cos 2x. dx`
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`
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
Integrate the function in x sin x.
Integrate the function in x (log x)2.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
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 : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int 1/sqrt(2x^2 - 5) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int logx/(1 + logx)^2 "d"x`
∫ log x · (log x + 2) dx = ?
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` 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.
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
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((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
`intsqrt(1+x) dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`int (logx)^2 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 the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3e^(x^2) dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int x^3 e^(x^2) dx`
