Advertisements
Advertisements
Question
Integrate the function:
`tan^(-1) sqrt((1-x)/(1+x))`
Advertisements
Solution
Let `I = int tan^-1 sqrt ((1 - x)/(1 + x)) dx`
Let x = cos θ
⇒ dx = -sinθ dθ
`= I = int tan^-1 sqrt ((1 - cos theta)/(1 + cos theta)) - sin theta d theta`
`= int - tan^-1 (tan theta/2) (sin theta) d theta`
`= - int theta/2 sin theta d theta`
`= -1/2 [theta int sin theta d theta - int d/(d theta) (theta) int sin theta d theta] d theta`
`= -1/2 [theta (- cos theta) - int 1 (-cos theta) d theta]`
`= 1/2 theta cos theta - 1/2 int cos theta d theta`
`= 1/2theta cos theta - 1/2 sin theta + C`
`= 1/2 theta cos theta - 1/2 sqrt (1 - cos^2 theta) + C`
`= 1/2 [x cos^-1 x sqrt (1 - x^2)] + C`
APPEARS IN
RELATED QUESTIONS
Find an anti derivative (or integral) of the following function by the method of inspection.
sin 2x
Find an anti derivative (or integral) of the following function by the method of inspection.
(axe + b)2
Find the following integrals:
`int (ax^2 + bx + c) dx`
Find the following integrals:
`int(2x^2 + e^x)dx`
Find the following integrals:
`int (x^3 + 5x^2 -4)/x^2 dx`
Find the following integrals:
`int (x^3 + 3x + 4)/sqrtx dx`
Find the following integrals:
`int (x^3 - x^2 + x - 1)/(x - 1) dx`
Find the following integrals:
`intsqrtx( 3x^2 + 2x + 3) dx`
Find the following integrals:
`intsec x (sec x + tan x) dx`
The anti derivative of `(sqrtx + 1/ sqrtx)` equals:
If `d/dx f(x) = 4x^3 - 3/x^4` such that f(2) = 0, then f(x) is ______.
Integrate the function:
`1/(x - x^3)`
Integrate the function:
`1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]`
Integrate the function:
`1/(x^2(x^4 + 1)^(3/4))`
Integrate the function:
`x^3/(sqrt(1-x^8)`
Integrate the function:
`cos^3 xe^(log sinx)`
Integrate the function:
`1/sqrt(sin^3 x sin(x + alpha))`
Integrate the functions `(sin^(-1) sqrtx - cos^(-1) sqrtx)/ (sin^(-1) sqrtx + cos^(-1) sqrtx) , x in [0,1]`
Integrate the function:
`(x^2 + x + 1)/((x + 1)^2 (x + 2))`
Evaluate `int tan^(-1) sqrtx dx`
Find : \[\int\frac{\left( x^2 + 1 \right)\left( x^2 + 4 \right)}{\left( x^2 + 3 \right)\left( x^2 - 5 \right)}dx\] .
Evaluate: `int (1 - cos x)/(cos x(1 + cos x)) dx`
If `d/(dx) f(x) = 4x^3 - 3/x^4`, such that `f(2) = 0`, then `f(x)` is
`sqrt((10x^9 + 10^x log e^10)/(x^10 + 10^x)) dx` equals
`int (e^x (1 + x))/(cos^2 (xe^x)) dx` equal
`int (dx)/sqrt(9x - 4x^2)` equals
`int (xdx)/((x - 1)(x - 2))` equals
`int x^2 e^(x^3) dx` equals
`int sqrt(1 + x^2) dx` is equal to
What is anti derivative of `e^(2x)`
If the normal to the curve y(x) = `int_0^x(2t^2 - 15t + 10)dt` at a point (a, b) is parallel to the line x + 3y = –5, a > 1, then the value of |a + 6b| is equal to ______.
`int (dx)/sqrt(5x - 6 - x^2)` equals ______.
