Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t.x : e2x sin x cos x
Advertisements
उत्तर
Let I = `int e^(2x)*sin x cos x*dx`
= `(1)/(2) int e(2x)*2sin x cos x dx`
= `(1)/(2) int e^(2x)*sin2x *dx` ...(1)
= `(1)/(2)[e^(2x) int sin 2x*dx - int {d/dx (e^(2x)) int sin 2x*dx}*dx]`
= `(1)/(2)[e(2x) ((-cos2x)/2) - int e^(2x) xx 2 xx ((- cos2x)/2)*dx]`
= `-(1)/(4) e^(2x) cos 2x + 1/2 int e^(2x) cos 2x*dx`
= `-(1)/(4)e^(2x) cos2x + (1)/(2)[e^(2x) int cos 2x*dx - int {d/dx (e^(2x)) int cos 2x*dx }*dx]`
= `(1)/(4)e^(2x) cos 2x + 1/2 [e^(2x).(sin2x)/(2) - int e^(2x) xx 2 xx (sin2x)/(2)*dx]`
= `-(1)/(4) e^(2x) cos 2x + (1)/(4) e^(2x) sin 2x - (1)/(2) int e^(2x) sin2x*dx`
∴ I = `-(1)/(4) e^(2x) cos 2x + (1)/(4) e^(2x) sin 2x - "I"` ..[By (1)]
∴ 2I = `-(1)/(4)e^(2x) cos 2x + 1/4e^(2x) sin2x`
∴ I = `e^(2x)/(8)(sin2x - cos2x) + c`.
APPEARS IN
संबंधित प्रश्न
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 cos-1 x.
Integrate the function in (sin-1x)2.
`int e^x sec x (1 + tan x) dx` equals:
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^2*cos^-1 x*dx`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
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))`
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
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
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int ("d"x)/(x - x^2)` = ______
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`int(1-x)^-2 dx` = ______
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
`int(xe^x)/((1+x)^2) dx` = ______
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int (logx)^2 dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
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 the following.
`intx^3 e^(x^2) dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate:
`int x^2 cos x dx`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
Which function is an example under priority \(I\) in the LIATE rule?
