Final answer:
To find the integral ∫√x (a² + x²)^(3/2) dx, we can use a substitution method. Let's substitute a² + x² = u. Then, differentiate both sides with respect to x. Rearrange the equation to solve for dx. Now, substitute the values of u and dx into the integral.
Step-by-step explanation:
To find the integral ∫√x (a² + x²)^(3/2) dx, we can use a substitution method. Let's substitute a² + x² = u. Then, differentiate both sides with respect to x:
2x dx = du
Rearrange the equation to solve for dx:
dx = (1/2x) du
Now, substitute the values of u and dx into the integral:
∫√x (a² + x²)^(3/2) dx = ∫√x u^(3/2) (1/2x) du
Simplify the expression:
∫√x (a² + x²)^(3/2) dx = (1/2) ∫u^(3/2) du
Integrate u^(3/2) using the power rule:
∫u^(3/2) du = (2/5) u^(5/2)
Substitute the value of u back into the equation:
(1/2) ∫u^(3/2) du = (1/2) (2/5) (a² + x²)^(5/2) + C
Therefore, the integral of √x (a² + x²)^(3/2) dx is (1/5) (a² + x²)^(5/2) + C, where C is the constant of integration.