Why is that integral written as the product of 2 other integrals?

Question

I am a student and just learning about integrals. **Could you please explain to me how the highlighted part of the equation was derived?**
I can't quite understand how was the c/a term brought of the integral and how 1 integral just became a product of 2 other ones.
Thank you in advance!

Welcome to MSE. It is in your best interest that you type your posts (using MathJax) instead of posting links to pictures. — José Carlos Santos, May 29 '22 at 11:11

José Carlos Santos · Accepted Answer · 2022-05-29T11:18:38.700

2

There is no product of two integrals there. What the author is claiming is that, since we always have$$\frac{cx+d}{ax+b}=\frac ca+\frac{d-\frac{bc}a}{ax+b},$$then$$\int\frac{cx+d}{ax+b}\,\mathrm dx$$is the sum of$$\int\frac ca\,\mathrm dx\quad\text{with}\quad\int\frac{d-\frac{bc}a}{ax+b}\,\mathrm dx.$$That is, it is the sum of$$\frac ca\int\mathrm dx\quad\text{with}\quad\left(d-\frac{bc}a\right)\int\frac{\mathrm dx}{ax+b}.$$

edited May 29 '22 at 11:18

answered May 29 '22 at 11:16

José Carlos Santos

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Yeah... Thanks a lot. That notaion used in a book was a little confusing to me, but now I get it. – Butyl May 29 '22 at 11:18
1

You left $\frac{c}{a}$ in the second Integral – jjagmath May 29 '22 at 11:18
@jjagmath I've edited my answer. Thank you. – José Carlos Santos May 29 '22 at 11:18
Welcome to MSE. This question goes against our quality standard policy. Instead of posting an answer, you should encourage the person who posted the question to improve it. – amWhy Jun 13 '22 at 15:49

Why is that integral written as the product of 2 other integrals?

1 Answers1