I am trying to understand the example in my book, however can't seem to makc sense of it. If someone could just explain it to me.

\(\displaystyle \frac{d}{dx}(3x-2)^{5} = 15(3x-2)^{4} \)

\(\displaystyle \int 15(3x-2)^{4} dx = (3x-2)^{5} + C \)

the next line I don't get, why do they divide both sides by 15? What's the need?

the book says 'an adjustment factor of 1/15 is needed'.

\(\displaystyle \int (3x-2)^{4} dx = \frac{1}{15}(3x-2)^{5} + c \)

If evaluating the above line only, Tweety...

\(\displaystyle \int{(3x-2)^4}dx=\frac{1}{15}(3x-2)^5+C\)

because of \(\displaystyle \frac{d}{dx}\left[(3x-2)^5+C\right]=15(3x-2)^4\)

hence \(\displaystyle \frac{1}{15}\ \frac{d}{dx}\left[(3x-2)^5+C\right]=(3x-2)^4\)

therefore \(\displaystyle \int{(3x-2)^4}dx=\frac{1}{15}\left[(3x-2)^5+C\right]\)

You can try it using substitution of course... \(\displaystyle u=3x-2,\ du=3dx,\ \frac{du}{3}=dx\)

\(\displaystyle \int{u^4}\frac{du}{3}=\frac{1}{3}\int{u^4}du=\frac{1}{3}\ \frac{1}{5}\ u^5+c=\frac{1}{15}(3x-2)^5+C\)