Advertisements
Advertisements
प्रश्न
Integrate the function in x cos-1 x.
Advertisements
उत्तर
Let `I = int x cos^-1 x dx = int cos^-1 x*x dx`
`= cos^-1 x* int x dx - int [d/dx (cos^-1 x) int x dx] dx`
`= cos^-1 x (x^2/2) - int (-1)/ sqrt (1 - x^2) (x^2/2) dx`
`= x^2/2 cos^-1 x + 1/2 int x^2/ sqrt (1 - x^2) dx`
∴ `I = x^2/2 cos^-1 x+ 1/2 I_1` ....(i)
Where `I_1 = int x^2/ sqrt (1 - x^2) dx`
Put x = cos θ
⇒ dx = -sinθ dθ
∴ `I_1 = int (cos^2 theta (-sin theta))/sqrt (1 - cos^2 theta) d theta`
`= - int cos^2 theta d theta = - 1/2 int (1 + cos 2 theta) d theta`
`= -1/2 (theta + (sin 2 theta)/2) + C`
`= -1/2 (theta + 1/2 xx 2 sin theta cos theta) + C`
`= - 1/2 (theta + cos theta sqrt (1 - cos^2 theta)) + C`
`= - 1/2 (cos^-1 x + x sqrt (1 - x^2)) + C` ....(ii)
From (i) and (ii), we get
`I = (2x^2 - 1) (cos^-1 x)/4 - x/4 sqrt (1 - x^2) + C`
APPEARS IN
संबंधित प्रश्न
Integrate the function in x sin 3x.
Integrate the function in x sec2 x.
Integrate the function in `(xe^x)/(1+x)^2`.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : log (log x)+(log x)–2
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"` =
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
`int 1/(4x + 5x^(-11)) "d"x`
`int sin4x cos3x "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int(x + 1/x)^3 dx` = ______.
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
`int tan^-1 sqrt(x) "d"x` is equal to ______.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`inte^x sinx dx`
Evaluate the following.
`intx^3 e^(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).
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
The value of `inta^x.e^x dx` equals
If \(u\) and \(v\) are differentiable functions, which formula is used for integration by parts?
Which function is an example under priority \(I\) in the LIATE rule?
