https://youtu.be/9xcwXVdH2ig

In this video I go over another example on integrating rational functions using partial fractions and this time show how sometimes even if the function we are integrating is a non-rational function we may be able to rationalize it or make it into a rational function using an appropriate substitution and thus be able to use partial fractions. The example I cover in this video is the integral of the function sqrt(x+4)/x and show how we can use the substitution u = sqrt(x+4) to transform the non-rational function into a rational function. This is a very useful technique to have when integrating non-rational functions so make sure to watch this video!


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# Integration of Rational Functions by Partial Fractions: Example 9: Rationalizing Substitutions

![Integration by Partial Fractions Example 9.jpeg](https://cdn.steemitimages.com/DQmYqtVDqSgryJoAp5Kb5FDPm8F7aWPnHEAYozu2R4g6nZ1/Integration%20by%20Partial%20Fractions%20Example%209.jpeg)

Some nonrational functions can be changed into rational functions by appropriate substitutions.

In particular, when the integrand contains an expression of the form:

![](https://cdn.steemitimages.com/DQmUmMv6YcybdBwNu72nbqcLdDXJJZh9WrgQSWcwEPkoRAh/image.png)

Then the following substitution may be effective:

![](https://cdn.steemitimages.com/DQmcqWr7Hn1cHrwnqaD1MLs45PfrUKsz1VCU2s2UHNyJKR4/image.png)

**Example:**

![](https://cdn.steemitimages.com/DQmZD2FYGvFhez9Ug7S43CqJrp6Je8fgCr6MPo4QghVqYDE/image.png)

**Solution:**

![](https://cdn.steemitimages.com/DQmeTLJUHPi5rNKdJ88i1KrQDd3atxAcJwhHFcxUSDAqpPx/image.png)

![](https://cdn.steemitimages.com/DQmZGXrz5hiBr9aPB92xz4gRpnQ42tDLF3WcAxLwpAuv1vj/image.png)

![](https://cdn.steemitimages.com/DQmYQ8v9xVqpD4G6G9xRGrk2MvLtDCnn6dNRQ4tGiqykSZo/image.png)