Advertisements
Advertisements
प्रश्न
Evaluate the following : `int x^2*cos^-1 x*dx`
उत्तर
Let I = `int x^2.cos^-1 x*dx`
= `int (cos^-1x)*x^2dx`
= `(cos^-1x) int x^2*dx- int d/dx(cos^-1x) int x^2*dx]*dx`
= `(cos^-1x) (x^3/3) - int ((-1)/sqrt(1 - x^2)) (x^3/3)*dx`
= `x^3/(3) cos^-1x + (1)/(3) int (x^2.x)/sqrt(1 - x^2)*dx`
In `int x^3/sqrt(1 - x^2)*dx`, put 1 – x2 = t
∴ – 2x.dx= dt
∴ x.dx = `-(1)/(2)dt`
Also, x2 = 1 – t
∴ I = `x^3/(3) cos^-1x + (1)/(3) int ((1 - t))/sqrt(t) (-1/2)*dt`
= `x^3/(3) cos^-1x - (1)/(6) int (1/sqrt(t) - sqrt(t))*dt`
= `x^3/(3) cos^-1x - (1)/(6) int t^(-1/2) dt + (1)/(6) int t^(1/2)*dt`
= `x^3/(3) cos^-1x - (1)/(6) (t^(1/2)/(1/2)) + (1)/(6) t^(3/2)/(3/2) + c`
= `x^3/(3) cos^-1x - (1)/(3)sqrt(1 - x^2) + (1)/(9)(1 - x^2)^(3/2) + c`.
APPEARS IN
संबंधित प्रश्न
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`
Integrate the function in x sin x.
Integrate the function in x log x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in x sec2 x.
Integrate the function in `e^x (1/x - 1/x^2)`.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following : `int x tan^-1 x .dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
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 (log (3x))/(xlog (9x))*dx` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(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) + cos (log x)]*dx` =
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "x"^2 "e"^"3x"`dx
`int ("x" + 1/"x")^3 "dx"` = ______
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
`int 1/(4x + 5x^(-11)) "d"x`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int 1/x "d"x` = ______ + c
Evaluate `int 1/(4x^2 - 1) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
`int cot "x".log [log (sin "x")] "dx"` = ____________.
Evaluate the following:
`int_0^1 x log(1 + 2x) "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
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
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 ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
`int1/sqrt(x^2 - a^2) dx` = ______
`intsqrt(1+x) 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
`inte^(xloga).e^x dx` is ______
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`intcos^-1(sqrt(x))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 tan^-1x dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))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 e^(x^2) dx`
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
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:
`inte^x "cosec" x(1 - cot x)dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
