Question

Integration using the substitution rule

Answer 1

\[\int\limits_{}^{} x(7+x^2)^{10} dx\]
u=7+x^2
du=2x
dx=du/2x=1/2x du

Answer 2

So it looks like you're only have a little trouble with the \(du\) part.
So check this out.\[\large u=7+x^2\]\[\large du=2x \;dx\]Dividing both sides by 2 gives us,\[\large \frac{1}{2}du=x\;dx\]

So what we have is,\[\large \color{orangered}{u=7+x^2}, \qquad \qquad \color{cornflowerblue}{\frac{1}{2}du=x\;dx}\]And we'll plug them into this,\[\large \int\limits\limits (\color{orangered}{7+x^2})^{10} \color{cornflowerblue}{x\;dx}\]

Answer 3

Do you see how those will plug in? :)
It's a little tricky to get used to.
You needed to solve for (x dx) since that's what was in your integral.