Advertisements
Advertisements
प्रश्न
Evaluate the following : `int cos(root(3)(x)).dx`
Advertisements
उत्तर
Let I = `int cos(root(3)(x)).dx`
Put `root(3)(x)` = t
∴ x = t3
∴ dx = 3t2.dt
∴ I = `int 3t^2 cos t.dt`
= `3t^2 int cos t.dt - int [d/dt (3t)^2 int cos t.dt].dt`
= `3t^2 sint - int 6t sint.dt`
= `3t^2 sint - [6t sin t.dt - int {d/dt (6t) int sin t.dt }.dt]`
= `3t^2 sint - [6t (- cos t) - int 6( - cos t).dt]`
= 3t2 sin t + 6t cos t – 6 sin t + c
= 3(t2 – 2) sin t + 6t cos t + c
= `3(x^(2/3) - 2) sin(root(3)(x)) + 6root(3)(x) cos(root(3)(x)) + c`.
APPEARS IN
संबंधित प्रश्न
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Integrate the function in x cos-1 x.
Integrate the function in tan-1 x.
Integrate the function in e2x sin x.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following : `int x.sin^2x.dx`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
∫ x log x dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
`int (sinx)/(1 + sin x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Evaluate `int 1/(x log x) "d"x`
Evaluate `int 1/(4x^2 - 1) "d"x`
`int logx/(1 + logx)^2 "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
`int 1/sqrt(x^2 - 9) dx` = ______.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
Solve: `int sqrt(4x^2 + 5)dx`
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int(1-x)^-2 dx` = ______
`intsqrt(1+x) dx` = ______
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
`inte^(xloga).e^x dx` is ______
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))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:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Which pair is listed as Exponential in the LIATE rule?
Using \[t=1-x^2,\] what is \[\int \frac{x\,dx}{\sqrt{1-x^2}}?\]
Evaluate \[\int x\cos x\,dx.\]
Evaluate \[\int e^x\sin x\,dx.\]
