A square root is the same as an exponential degree of 1/2, so a square root function can be integrated using the same formula for polynomials. A u-substitution for the expression under the square root symbol is a common additional step. Find the integral of square root functions by rewriting the square root as u^(1/2) and then finding the anti-derivative using the polynomial anti-derivative formula from calculus.
Perform a u-substitution by replacing the expression inside the square root with u. For example, replace the expression (3x - 5) in the function f(x) = 6√(3x - 5) to get the new function f(x) = 6√u.
Rewrite the square root as an exponential degree 1/2. For example, rewrite the function f(x) = 6√u + 2, as 6u^(1/2).
Calculate the derivative du/dx and isolate dx in the equation. In the above example, the derivative of u = 3x - 5 is du/dx = 3. Isolating dx yields the equation dx = (1/3) du.
Replace the dx in the integral expression with its value in terms of du, which you just did. Continuing the example, the integral of 6u^(1/2) dx becomes the integral of f(u) = 6u^(1/2) * (1/3) du, or 2u^(1/2) du.
Evaluate the anti-derivative of the function f(u) using the anti-derivative formula for a*x^n: a(x^(n + 1)) / (n + 1). In the above example, the anti-derivative of f(u) = 2u^(1/2) is 2(u^(3/2)) / (3/2), which simplifies to (4/3)u^(3/2).
Substitute the value of x back in for u to complete the integration. In the above example, substitute "3x - 5" back in for u to get the value of the integral in terms of x: F(x) = (4/3)(3x - 5)^(3/2).
Rewrite the expression in radical form, if you wish, by replacing the exponent (3/2) with a square root of the expression to the third power. In the above example, rewrite F(x) in radical form as F(x) = (4/3)√((3x - 5)^3).