Advertisements
Advertisements
Question
Integrate the function in e2x sin x.
Advertisements
Solution
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
RELATED QUESTIONS
Integrate the function in x sin 3x.
Integrate the function in (sin-1x)2.
Integrate the function in x sec2 x.
Integrate the function in x (log x)2.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*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 x^2 *e^(3x)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int (sinx)/(1 + sin x) "d"x`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int 1/x "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
`int logx/(1 + logx)^2 "d"x`
∫ log x · (log x + 2) dx = ?
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
`int_0^1 x tan^-1 x dx` = ______.
`intsqrt(1+x) dx` = ______
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`int (logx)^2 dx`
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
