Advertisements
Advertisements
Question
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Advertisements
Solution
Let, I = `int e^x ((1 - sinx)/(1 - cosx))dx`
= `int e^x (1/(1 - cosx) - sinx/(1 - cosx))dx`
= `int e^x ((-sinx)/(1 - cosx) + 1/(1 - cosx))dx`
Let, f(x) = `(-sinx)/(1 - cosx)`
f'(x) = `-[(cosx(1 - cosx) - sinx (sinx))/(1 - cosx)^2]`
= `-[(cosx - cos^2x - sin^2x)/(1 - cosx)^2]`
= `-[(cosx - (cos^2x + sin^2x))/(1 - cosx)^2]`
= `-[(cosx - 1)/(1 - cosx)^2]`
= `(1 - cosx)/(1 - cosx)^2`
= `1/(1 - cosx)`
Hence, the given integration is of form
`int e^x [f(x) + f^'(x)]dx = e^x f(x)`
where f(x) = `(-sinx)/(1 - cosx)` and f'(x) = `1/(1 - cosx)`
∴ I = `e^x xx ((-sinx)/(1 - cosx))`
= `(e^x sinx)/((cosx - 1))`.
APPEARS IN
RELATED QUESTIONS
Integrate : sec3 x w. r. t. x.
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
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x cos-1 x.
Integrate the function in ex (sinx + cosx).
Integrate the function in `(xe^x)/(1+x)^2`.
Integrate the function in `e^x (1/x - 1/x^2)`.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x.cos^3x.dx`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
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
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"/(3 - 2"x" - "x"^2)`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int cot "x".log [log (sin "x")] "dx"` = ____________.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Find: `int e^x.sin2xdx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int_0^1 x tan^-1 x dx` = ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
`int1/(x+sqrt(x)) 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 (1 + x + x^2/(2!))dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
