How to Integrate Square Root Functions

By Karl Wallulis
The integral of a function gives the value of the area under the curve of its corresponding graph.

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).

About the Author

Karl Wallulis has been writing since 2010. He has written for the Guide to Online Schools website, covering academic and professional topics for young adults looking at higher-education opportunities. Wallulis holds a Bachelor of Arts in psychology from Whitman College.