Advertisements
Advertisements
प्रश्न
Evaluate the following : `int e^(2x).cos 3x.dx`
Advertisements
उत्तर
Let I = `int e^(2x).cos 3x.dx`
I = `int cos 3x.e^(2x) dx`
= `cos 3x inte^(2x) .dx - int [d/dx (cos 3x) - e^(2x).dx]dx`
= `cos3x. (e^(2x))/(2) - int(-sin3x).(3) e^(2x)/2.dx`
= `(1)/(2).cos3xe^(2x) + 3/2 int sin 3x. e^(2x) dx`
= `(1)/(2)cos3xe^(2x) + 3/2[sin3x.int e^(2x)dx - int [(cos3x)3.int e^(2x)dx]dx`
= `(1)/(2)cos3x.e^(2x) + 3/2sin3x.(e^(2x))/2 - 3/2 .3int cos3x.e^(2x)/2dx`
= `(1)/(2)cos3x.e^(2x) + 3/4sin3x.e^(2x) - 9/4 intcos3x.e^(2x)dx`
= `(1)/(2)cos3x.e^(2x) + 3/4sin3x.e^(2x) - 9/4 "I"`
`"I" + 9/4"I" = (1/2 cos3x + 3/4 sin3x)e^(2x)`
`13/4"I" = (1/2 cos3x + 3/4 sin3x)e^(2x)`
I = `4/13 [1/2cos3x + 3/4sin3x]e^(2x)`
I = `1/13 [2cos3x + 3sin3x]e^(2x) + c`
∴ I = `e^(2x)/(13) (2 cos3x + 3 sin 3x) + c`.
APPEARS IN
संबंधित प्रश्न
Integrate the function in x log 2x.
Integrate the function in x2 log x.
Integrate the function in x cos-1 x.
Integrate the function in `(xe^x)/(1+x)^2`.
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 sec^3x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Choose the correct options from the given alternatives :
`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
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 tan(sin^-1 x)*dx` =
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` 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 "dx"/("9x"^2 - 25)`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
Find `int_0^1 x(tan^-1x) "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 1/sqrt(x^2 - 9) dx` = ______.
Solve: `int sqrt(4x^2 + 5)dx`
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Solution of the equation `xdy/dx=y log y` is ______
`inte^(xloga).e^x dx` is ______
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`int (logx)^2 dx`
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`intx^3 e^(x^2)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
`∫ sin^(−1)` xdx is equal to ______.
