How To Integrate Square Root Functions
Integrating functions is one of the core applications of calculus. Sometimes, this is straightforward, as in:
\(F(x) = \int( x^3 + 8) dx\)
In a comparatively complicated example of this type, you can use a version of the basic formula for integrating indefinite integrals:
\(\int (x^n + A) dx = \frac{x^{(n + 1)}}{n + 1} + Ax + C\)
where A and C are constants.
Thus for this example,
\(\int x^3 + 8 = \frac{x^4}{4} + 8x + C\)
Integration of Basic Square Root Functions
On the surface, integrating a square root function is awkward. For example, you may be stymied by:
\(F(x) = \int \sqrt{(x^3) + 2x – 7}dx\)
But you can express a square root as an exponent, 1/2:
\(\sqrt{x^3} = x^{3(1/2)} = x^{(3/2)}\)
The integral therefore becomes:
\(\int (x^{3/2} + 2x – 7)dx\)
to which you can apply the usual formula from above:
\(\begin{aligned}
\int (x^{3/2} + 2x – 7)dx &= \frac{x^{(5/2)}}{5/2} + 2\bigg(\frac{x^2}{2}\bigg) – 7x\)
\(&= \frac{2}{5}x^{(5/2)} + x^2 – 7x
\end{aligned}\)
Integration of More Complex Square Root Functions
Sometimes, you may have more than one term under the radical sign, as in this example:
\(F(x) = \int \frac{x + 1}{\sqrt{x – 3}}dx\)
You can use u-substitution to proceed. Here, you set u equal to the quantity in the denominator:
\(u = \sqrt{x – 3}\)
Solve this for x by squaring both sides and subtracting:
\(u^2 = x – 3\)
\(x = u^2 + 3\)
This allows you to get dx in terms of u by taking the derivative of x:
\(dx = (2u)du\)
Substituting back into the original integral gives
\(\begin{aligned}\)
\(F(x) &= \int \frac{u^2 + 3 + 1}{u}(2u)du\)
\(&= \int \frac{2u^3 + 6u + 2u}{u}du\)
\(&= \int (2u^2 + 8)du\)
\(\end{aligned}\)
Now you can integrate this using the basic formula and expressing u in terms of x:
\(\begin{aligned}\)
\(\int (2u^2 + 8)du &= \frac{2}{3}u^3 + 8u + C\)
\(&= \frac{2}{3} (\sqrt{x – 3})^3 + 8( \sqrt{x – 3}) + C\)
\(&= \frac{2}{3} (x – 3)^{(3/2)} + 8(x – 3)^{(1/2)} + C\)
\(\end{aligned}\)
Cite This Article
MLA
Beck, Kevin. "How To Integrate Square Root Functions" sciencing.com, https://www.sciencing.com/integrate-square-root-functions-8625260/. 14 December 2020.
APA
Beck, Kevin. (2020, December 14). How To Integrate Square Root Functions. sciencing.com. Retrieved from https://www.sciencing.com/integrate-square-root-functions-8625260/
Chicago
Beck, Kevin. How To Integrate Square Root Functions last modified August 30, 2022. https://www.sciencing.com/integrate-square-root-functions-8625260/