how do you intergrate
sqrt((1+((1/4)/(4-x^2)))
To integrate the given expression, we will break it down into simpler parts and use the basic rules of integration. Here's how you can integrate:
1. Start by expanding the expression inside the square root if possible. In this case, there isn't any simplification possible.
2. Rewrite the expression using fractional exponents. Square roots can be expressed as fractional exponents. The expression can be written as:
√(1 + ((1/4) / (4 - x^2))) = (1 + ((1/4) / (4 - x^2)))^(1/2)
3. Now, we can integrate by applying the power rule of integration. According to the power rule, the integral of x^n is (x^(n+1)) / (n+1).
Integral of (1 + ((1/4) / (4 - x^2)))^(1/2) dx
4. Let's make a substitution to simplify the integration. Let u = 4 - x^2. Then, du = -2x dx.
The expression becomes:
Integral of (1 + ((1/4) / u))^(1/2) (-1/2x) du
5. Simplify the expression further:
(-1/2) Integral of (1 + ((1/4) / u))^ (1/2) du
6. Now, let's integrate using the power rule. Add 1 to the exponent and divide by the new exponent:
(-1/2) [(2u + 1) / (2(1/2))] + C
Simplifying further:
-(u + 1/2) + C
7. Substitute back the value of u:
-(4 - x^2 + 1/2) + C
8. Finally, simplify the expression:
-(7/2 - x^2) + C
Therefore, the integral of sqrt((1 + ((1/4) / (4 - x^2))) dx is -(7/2 - x^2) + C, where C is the constant of integration.