Advertisements
Advertisements
प्रश्न
Integrate the function in e2x sin x.
Advertisements
उत्तर
Let `I = inte^(2x) sinx dx`
`= e^(2x) int sin x dx - int [d/dx (e^(2x))* int sin x dx] dx`
`= e^(2x) (- cos x) - int 2e^(2x) (- cos x) dx + C_1`
`= -e^(2x) cos x + 2 int e^(2x) cos x dx + C_1`
`= -e^(2x) cos x + 2` `...[e^(2x) int cos x dx - int (d/dx (e^(2x))* int cos xdx) dx] + C_1`
`= -e^(2x) cos x + 2e^(2x) sin x - 4 int e^(2x) sin x dx + C_1 + C_2`
`= e^(2x) (2 sin x - cos x) - 4I + C_1 + C_2`
∵ `5I = e^(2x) (2 sinx - cos x) + C_1 + C_2`
⇒ `I = (e^(2x))/5 (2 sin x - cos x) + C`
where C = C1 + C2
APPEARS IN
संबंधित प्रश्न
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Integrate the function in (x2 + 1) log x.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int x.cos^3x.dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
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 sin (log x)*dx` =
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int [1/(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"/sqrt(4"x"^2 - 5)`
Evaluate: `int "dx"/("9x"^2 - 25)`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int 1/x "d"x` = ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
`int tan^-1 sqrt(x) "d"x` is equal to ______.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
`int(logx)^2dx` equals ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
`int logx dx = x(1+logx)+c`
Evaluate:
`inte^x sinx dx`
Evaluate:
`int e^(logcosx)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:
`int x^2 cos x dx`
