Advertisements
Advertisements
Question
If y = `(x sin^-1 x)/sqrt(1 -x^2)`, prove that: `(1 - x^2)dy/dx = x + y/x`
Solution
Here, we have y = `(x sin^-1 x)/sqrt(1 -x^2)`
y `sqrt(1 -x^2)` = x sin-1 x ....(i)
Differentiate both sides w.r.t. x, we have
`y ((-2x))/(2sqrt(1 -x^2)) + sqrt(1 - x^2) (dy)/(dx) = x (1)/sqrt(1 -x^2) + sin^-1 x`
`- xy + (1 - x^2) (dy)/(dx) = x + sqrt(1 - x^2) sin^-1 x`
`- xy + (1 - x^2) (dy)/(dx) = x + sqrt(1 - x^2) . (y)/(x) sqrt(1 -x^2) ...[ ∵ sin^-1 x = (y)/(x) sqrt(1 - x^2) , "using" (i) ]`
`- xy + (1 - x^2) (dy)/(dx) = x + (y)/(x) (1 - x^2)`
`- xy + (1 - x^2) (dy)/(dx) = x + (y)/(x) - yx`
`(1 - x^2) (dy)/(dx) = x + (y)/(x) `
APPEARS IN
RELATED QUESTIONS
Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4) `
If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.
Prove the following:
`3cos^(-1) x = cos^(-1)(4x^3 - 3x), x in [1/2, 1]`
if `tan^(-1) (x-1)/(x - 2) + tan^(-1) (x + 1)/(x + 2) = pi/4` then find the value of x.
Find the value of the given expression.
`sin^(-1) (sin (2pi)/3)`
Find the value of the given expression.
`tan^(-1) (tan (3pi)/4)`
Solve the following equation for x: `cos (tan^(-1) x) = sin (cot^(-1) 3/4)`
Solve for x : \[\tan^{- 1} \left( \frac{x - 2}{x - 1} \right) + \tan^{- 1} \left( \frac{x + 2}{x + 1} \right) = \frac{\pi}{4}\] .
Solve: tan-1 4 x + tan-1 6x `= π/(4)`.
Find the value of `cot[sin^-1 3/5 + sin^-1 4/5]`
Choose the correct alternative:
If `sin^-1x + cot^-1 (1/2) = pi/2`, then x is equal to
Evaluate: `tan^-1 sqrt(3) - sec^-1(-2)`.
Prove that `2sin^-1 3/5 - tan^-1 17/31 = pi/4`
Prove that cot–17 + cot–18 + cot–118 = cot–13
The value of the expression `tan (1/2 cos^-1 2/sqrt(5))` is ______.
If |x| ≤ 1, then `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` is equal to ______.
If cos–1α + cos–1β + cos–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals ______.
The value of cot–1(–x) for all x ∈ R in terms of cot–1x is ______.
The value of cot `("cosec"^-1 5/3 + "tan"^-1 2/3)` is ____________.
If `"tan"^-1 2 "x + tan"^-1 3 "x" = pi/4`, then x is ____________.
`"cos"^-1["cos"(2"cot"^-1(sqrt2 - 1))]` = ____________.
If `6"sin"^-1 ("x"^2 - 6"x" + 8.5) = pi,` then x is equal to ____________.
`"tan"^-1 (sqrt3)`
If `"sin" {"sin"^-1 (1/2) + "cos"^-1 "x"} = 1`, then the value of x is ____________.
Find the value of `cos^-1 (1/2) + 2sin^-1 (1/2) ->`:-
`tan(2tan^-1 1/5 + sec^-1 sqrt(5)/2 + 2tan^-1 1/8)` is equal to ______.
The set of all values of k for which (tan–1 x)3 + (cot–1 x)3 = kπ3, x ∈ R, is the internal ______.
If `tan^-1 ((x - 1)/(x + 1)) + tan^-1 ((2x - 1)/(2x + 1)) = tan^-1 (23/36)` = then prove that 24x2 – 23x – 12 = 0
