https://youtu.be/ql2mu5joNJI

In this video I go over another example on integration of rational functions by partial fractions and this time integrate a function that has a non-reducible non-linear factor in the denominator. This requires applying the method of partial fraction decomposition but for non-linear factors as I have shown in my earlier videos. The example I go over is the integral of (2x<sup>2</sup> - x + 4)/(x<sup>3</sup> + 4x). In the example I derive the integral of 1/(x<sup>2</sup> + a<sup>2</sup>) = 1/a * inverse tan (x/a) which was required in solving the overall integral.

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# Integration of Rational Functions by Partial Fractions: Example 5

![Integration by Partial Fractions Example 5.jpeg](https://cdn.steemitimages.com/DQmcVgKFX9p4tCkjjF5woesN2SWgwR2r4qGANtegEcX8HMV/Integration%20by%20Partial%20Fractions%20Example%205.jpeg)

**Example:**

![](https://cdn.steemitimages.com/DQmRcVfj6gaszvJEVgQsvuoVqRxWdRFCLey8bBGRyFn69vv/image.png)

**Solution:**

![](https://cdn.steemitimages.com/DQmc4G5nffQhLFMrtr5zbgh2FVVGYoxFEn82vD7Yk2xzi8d/image.png)

![](https://cdn.steemitimages.com/DQmNw2btUXAinE6yPidCmefwH4WXyiFDBWN64AP6VoMsDr8/image.png)

![](https://cdn.steemitimages.com/DQmYLtaeL5uKqoXrn1MCEASfY52dPVgMLQEqQHqhqzpBAy9/image.png)