The purpose of integrating Sqrt[x^2-a] is to find the area under the curve of the function over a given interval. It is also used to calculate displacement, velocity, and acceleration in physics and engineering problems.
The general approach to integrating Sqrt[x^2-a] is to use a substitution method, where we assign a new variable to the expression inside the square root. This allows us to convert the integral into a more manageable form and solve it using basic integration techniques.
The appropriate substitution for integrating Sqrt[x^2-a] depends on the value of "a". If "a" is a positive number, we use x = a*sinh(u) or x = a*tanh(u) as the substitution. If "a" is a negative number, we use x = -a*cosh(u) or x = -a*tan(u) as the substitution. This allows us to eliminate the square root and simplify the integral.
The final answer to the integral of Sqrt[x^2-a] is (x/2)*Sqrt[x^2-a] + (a/2)*ln|x + Sqrt[x^2-a]| + C. This can be derived by using the appropriate substitution and then using basic integration techniques such as u-substitution and integration by parts.
Sure, let's say we want to integrate Sqrt[x^2-9]. Using the substitution x = 3*sinh(u), we can rewrite the integral as 3*sinh(u)*cosh(u)du, which simplifies to (9/2)*sinh(2u) + C. Substituting back for x, the final answer is (3x/2)*Sqrt[x^2-9] + (9/2)*ln|x + Sqrt[x^2-9]| + C.