Advertisements
Advertisements
प्रश्न
Evaluate the following : `int sin θ.log (cos θ).dθ`
Advertisements
उत्तर
Let I = `int sin θ.log (cos θ).dθ`
= `int log(cosθ).sinθ dθ`
Put cos θ = t
∴ – sin θ dθ = dt
∴ sin θ dθ = – dt
∴ I = `int logt.(- dt)`
= `- int (logt).1dt`
= `-[(log t) int 1dt - int {d/dt (log t) int 1 dt }dt]`
= `-[(logt)t - int1/t.t dt]`
= `- t logt + int 1 dt`
= – t log t + t + c
= – cos θ . log (cos θ) + cos θ + c
= – cos θ [log (cos θ) – 1] + c.
APPEARS IN
संबंधित प्रश्न
Integrate the function in x cos-1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in (x2 + 1) log x.
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*cos^-1 x*dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Evaluate the following.
∫ x log x dx
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int(x + 1/x)^3 dx` = ______.
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int"e"^(4x - 3) "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int 1/(x log x) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
`int_0^1 x tan^-1 x dx` = ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int1/sqrt(x^2 - a^2) dx` = ______
`intsqrt(1+x) dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int1/(x+sqrt(x)) dx` = ______
`inte^(xloga).e^x dx` is ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate `int tan^-1x 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 the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(1 + x + x^2/(2!))dx`.
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
