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
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 (x2 + 1) log x.
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log 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` =
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : e2x sin x cos x
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int(x + 1/x)^3 dx` = ______.
`int 1/x "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int 1/(x(x - 1)) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
Solve: `int sqrt(4x^2 + 5)dx`
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.
Solution: (x2 + y2) dx - 2xy dy = 0
∴ `dy/dx=(x^2+y^2)/(2xy)` ...(1)
Puty = vx
∴ `dy/dx=square`
∴ equation (1) becomes
`x(dv)/dx = square`
∴ `square dv = dx/x`
On integrating, we get
`int(2v)/(1-v^2) dv =intdx/x`
∴ `-log|1-v^2|=log|x|+c_1`
∴ `log|x| + log|1-v^2|=logc ...["where" - c_1 = log c]`
∴ x(1 - v2) = c
By putting the value of v, the general solution of the D.E. is `square`= cx
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`int e^(logcosx)dx`
`int (sin^-1 sqrt(x) + cos^-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^3e^(x^2) dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^2e^(4x)dx`
The value of `inta^x.e^x dx` equals
Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3) dx`
The value of `int (x sin^-1)/(sqrt(1 - x^2)) dx` is equal to:
