Advertisements
Advertisements
प्रश्न
Evaluate the following : `int x^2*cos^-1 x*dx`
Advertisements
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(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Integrate the function in x sin 3x.
Integrate the function in `x^2e^x`.
Integrate the function in x sec2 x.
Integrate the function in ex (sinx + cosx).
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following: `int logx/x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
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` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "x"^2 *"e"^"3x"`dx
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
Evaluate:
∫ (log x)2 dx
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Evaluate `int 1/(x log x) "d"x`
`int logx/(1 + logx)^2 "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
`int 1/sqrt(x^2 - 9) dx` = ______.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
`int(1-x)^-2 dx` = ______
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate:
`int (logx)^2 dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate the following.
`intx^3e^(x^2) dx`
Evaluate `int (1 + x + x^2/(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`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
