Advertisements
Advertisements
Question
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Advertisements
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
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`
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x sin 3x.
Integrate the function in tan-1 x.
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
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 [sin (log x) + cos (log x)]*dx` =
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int (sinx)/(1 + sin x) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sin4x cos3x "d"x`
`int ("d"x)/(x - x^2)` = ______
Evaluate `int 1/(x(x - 1)) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Solve: `int sqrt(4x^2 + 5)dx`
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
Find `int e^x ((1 - sinx)/(1 - cosx))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 ______.
`int logx dx = x(1+logx)+c`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`inte^x sinx dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3e^(x^2) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`intx^2e^(4x)dx`
The value of `inta^x.e^x dx` equals
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
