Find the intergral of (1+x)/sqrt(3-x^2)
To find the integral of (1+x)/sqrt(3-x^2), we can use a technique called trigonometric substitution. Here's how you can do it:
Step 1: Identify the trigonometric substitution
Let's let x = sqrt(3)sin(theta), where -pi/2 <= theta <= pi/2. This substitution allows us to express sqrt(3-x^2) in terms of trigonometric functions.
Step 2: Compute dx in terms of d(theta)
To find dx in terms of d(theta), we can differentiate x = sqrt(3)sin(theta) with respect to theta.
dx/dtheta = sqrt(3)cos(theta)
Step 3: Substitute the integral and dx in terms of theta
Using the substitution x = sqrt(3)sin(theta), we substitute x and dx in the integral and get:
∫(1+x)/sqrt(3-x^2) dx = ∫[(1+sqrt(3)sin(theta))/sqrt(3-(sqrt(3)sin(theta))^2)] * sqrt(3)cos(theta) dθ
Step 4: Simplify the expression
Simplify the expression further:
∫(1+x)/sqrt(3-x^2) dx = ∫(1+√3sin(theta))/cos(theta) dθ
Step 5: Evaluate the integral
Now, we have a new integral in terms of theta. We can evaluate this integral using basic integration techniques. The integral of (1+√3sin(theta))/cos(theta) can be solved using trigonometric identities and the substitution u = tan(theta/2).
Step 6: Calculate the final result
After integrating, we get:
∫(1+x)/sqrt(3-x^2) dx = √3arcsin(x/√3) + C
So, the integral is √3arcsin(x/√3) + C, where C is the constant of integration.