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# Evaluate the integral.$\displaystyle \int \frac{x^4 + 3x^2 + 1}{x^5 + 5x^3 + 5x}\ dx$

## $$\int \frac{x^{4}+3 x^{2}+1}{x^{5}+5 x^{3}+5 x} d x=\frac{1}{5} \ln \left|x^{5}+5 x^{3}+5 x\right|+K$$

#### Topics

Integration Techniques

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### Video Transcript

let's evaluate to given in several here before we do partial fraction to composition. Let's just go ahead and try a new substitution, sometimes making the right choice of you before you do partial fraction. The composition will give you a much easier problem here. Let's go ahead and take the differential. Fifty. Next, plus five. So we have. Do you over five is basically is exactly equal to the numerator of the incident. So after our u substitution, we have won over you. Do you? And we also have a pull off his five here, Look. So in this case, we took a rough look infraction. Maybe you substitution. And now look at our partial fashion. We already had this one over you so we can go ahead and integrate this one over five natural law. Give you sort of five down there. Not a six. All right. Closer constancy of integration. And then the last step is just to go back to the use up and go from you back tax X to the fifth five x cubed and then five six. And that's our answer