Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Advertisements
Solution
Let I = `int e^x ((1 + sin x)/(1 + cos x)).dx`
= `int e^x [(1 + 2sin x/2 cos x /2)/(2 cos^2 x/2)].dx`
= `int e^x [(1)/(2cos^2 x/2) + (2sin x/2 cos x/2)/(2cos^2 x/2)].dx`
= `int e^x[1/2 sec^2 x/2 + tan (x/2)].dx`
Put f(x) = `tan (x/2)`
∴ f'(x) = `d/dx [tan x/2]`
= `sec^2 x/(2).(1)/(2)`
= `(1)/(2) sec^2 x/(2)`
∴ I = `int e^x [f(x) + f'(x)].dx`
= ex f(x) + c
= `e^x. tan (x/2) + c`.
APPEARS IN
RELATED QUESTIONS
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in tan-1 x.
Integrate the function in `(xe^x)/(1+x)^2`.
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : sec4x cosec2x
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
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"/(25"x" - "x"(log "x")^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
Evaluate `int 1/(x(x - 1)) "d"x`
Evaluate `int 1/(x log x) "d"x`
Evaluate `int 1/(4x^2 - 1) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
Find `int_0^1 x(tan^-1x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int_0^1 x tan^-1 x dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
`intsqrt(1+x) dx` = ______
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate:
`int (logx)^2 dx`
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`
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`
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))`
Evaluate `int (1 + x + x^2/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^2e^(4x)dx`
The value of `inta^x.e^x dx` equals
