Advertisements
Advertisements
प्रश्न
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
उत्तर
Let `I==int_0^pie^(2x)sin(pi/2+x)dx`
Integrating by parts, we get
` I=1/2[e^(2x)sin(pi/4+x)_0^pi]-1/2int_0^pie^(2x)cos(pi/4+x)dx`
Now, integrating the second term by parts, we get
` =>I=1/2[e^(2x)sin(pi/4+x)_0^pi]-1/2{[1/2e^(2x)cos(pi/4+x)_0^pi]+1/2int_0^pi e^(2x)sin(pi/4+x)dx}`
=>`I=1/2[e^(2x)sin(pi/4+x)_0^pi]-1/4[e^(2x)cos(pi/4+x)_0^pi]-1/4I`
`=>5/4I=1/2[e^(2x)sin(pi+pi/4)-sin(pi/4)]-1/4[e^(2x)cos(pi+pi/4)-cos(pi/4)]`
`=>5/4I=1/2 |__-e^(2x)xx1/sqrt2-1/sqrt2__|-1/4|__-e^(2pi)xx1/sqrt2-1/sqrt2__|`
`=>5/4I==1/(2sqrt2)e^(2pi)-1/(2sqrt2)+1/(4sqrt2)e^(2pi)+1/(4sqrt2)`
`=>I=-1/(5sqrt2)(e^(2pi)+1)`
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
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
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 x sin 3x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in tan-1 x.
Integrate the function in x (log x)2.
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
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 (log (3x))/(xlog (9x))*dx` =
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
`int(logx)^2dx` equals ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
`intsqrt(1+x) dx` = ______
`int(xe^x)/((1+x)^2) dx` = ______
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
