Advertisements
Advertisements
Question
Integrate the function in (sin-1x)2.
Advertisements
Solution
Let `I = int (sin^-1 x)^2 dx`
Put `sin^-1 x = theta`
⇒ x = sinθ
⇒ dx = cosθ dθ
∴ `I = int theta^2 cos theta d theta`
`= theta^2 int (cos theta) d theta - int (d/ (d theta) (theta^2) * int cos theta d theta) d theta`
`= theta^2 (sin theta) - int 2 theta (sin theta) d theta`
`= theta^2 sin theta - 2 int theta sin theta d theta + C`
`= theta^2 sin theta - 2 [theta * (- cos theta) - int 1 * (- cos theta) d theta] + C`
`= theta^2 sin theta + 2 theta cos theta - 2 int cos theta d theta + C`
`= theta^2 sin theta + 2 theta sqrt (1 - sin^2 theta) - 2 sin theta + C`
`= x (sin^-1 x)^2 + 2sin^-1 x sqrt (1 - x^2) - 2x + C`
APPEARS IN
RELATED QUESTIONS
Integrate the function in x tan-1 x.
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in x sec2 x.
Integrate the function in ex (sinx + cosx).
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Find :
`∫(log x)^2 dx`
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
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 (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` dx
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Evaluate `int 1/(x(x - 1)) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "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 - a^2)dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
`int(xe^x)/((1+x)^2) 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 the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate the following.
`intx^3e^(x^2) dx`
