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
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`
Integrate the function in x log 2x.
Integrate the function in (x2 + 1) log x.
Integrate the function in e2x sin x.
`intx^2 e^(x^3) dx` equals:
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
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 (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : e2x sin x cos x
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate:
∫ (log x)2 dx
`int (sinx)/(1 + sin x) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
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 x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Find: `int e^x.sin2xdx`
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
`int logx dx = x(1+logx)+c`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Which pair is listed as Trigonometric in the LIATE rule?
When applying integration by parts to \(\log x\) and inverse trig, what should they be multiplied by?
