Advertisements
Advertisements
Question
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Solution
Let I = `int [log(1 + cosx) - xtan(x/2)]*dx`
= `int [log(1 + cos.x)*1dx - intxtan (x/2)*dx`
= `[log(1 + cosx)]* int 1dx - int {d/dx [log (1 + cosx)]* int 1dx}*dx - xtan (x/2)*dx`
= `[log (1 + cosx)]*(x) - int 1/(1 + cosx)*(0 - sin x)*xdx - int x tan (x/2)*dx`
= `x*log(1 + cosx) + intx* (sinx)/(1 + cosx)*dx - int xtan (x/2)*dx + c`
= `x*log(1 + cosx) + intx*(2sin(x/2)*cos(x/2))/(2cos^2(x/2)*dx - int xtan (x/2)*dx + c`
= `xlog (1 + cosx) + int x*tan(x/2)*dx - intxtan(x/2)*dx + c`
= x·log(1 + cosx) + c.
APPEARS IN
RELATED QUESTIONS
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
Integrate the function in x sin x.
Integrate the function in x log 2x.
Integrate the function in tan-1 x.
Integrate the function in (x2 + 1) log 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)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Integrate the function in e2x sin x.
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 x^2.log x.dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int cos sqrt(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:
sin (log x)
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `log(1 + x)^((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` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : log (x2 + 1)
Evaluate the following.
`int "x"^2 "e"^"4x"`dx
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
Evaluate: ∫ (log x)2 dx
`int 1/(4x + 5x^(-11)) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sin4x cos3x "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int(x + 1/x)^3 dx` = ______.
`int"e"^(4x - 3) "d"x` = ______ + c
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
Evaluate `int 1/(x(x - 1)) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int cot "x".log [log (sin "x")] "dx"` = ____________.
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int_0^pi x log sin x "d"x`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
`int 1/sqrt(x^2 - 9) dx` = ______.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
`int(logx)^2dx` equals ______.
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int1/(x+sqrt(x)) dx` = ______
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`int e^(logcosx)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`
Evaluate:
`int1/(x^2 + 25)dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
