finding the integral with substitution

Question

I have to find the integral of a function and I am not sure about the beginning of the integral. I have solved it but I would like to know why the following procedure happens

$\int x^7 \sqrt{5+3x^4}dx = 1/4 \int u \sqrt{3u+5}du$

$u=x^4$

$du = 4x^3$

Should it be $x^7$ instead of $u$? Why is it $u$?

Thank you

Answer 1

After performing u-substitution with:

$$u = x^4$$

and

$$du = 4x^3dx$$

You simply have to plug in and simplify:

$$\int x^7 \sqrt{5+3x^4}dx = \frac{1}{4}\int x^4 \sqrt{3x^4+5}*(4x^3dx) = \frac{1}{4}\int u \sqrt{3u+5}du$$

Hope this helped.

Answer 2

I am not really sure this is what you're asking, but basically you have $x^7$ which is the same thing as $x^4 x^3$, so you have substituted $x^4$ with $u$ and $x^3dx$ with $\frac{1}{4}du$. Hope this helps.