Question

integral of 3 root 6-x

Answer 1

\[\large \int\limits 3\sqrt{6-x}\; dx\]Like this? :)

Answer 2

We can pull the constant `3` outside of the integral, it won't affect the integration process at all.\[\large 3\int\limits\limits \sqrt{6-x}\; dx\]From here we can apply a `U substitution`.
Are you comfortable with doing U substitutions yet?
This is a nice easy example to get familiar with the process.

Answer 3

Let \(\large \color{royalblue}{u=6-x}\),
Taking the derivative of our substitution, with respect to x, gives us,\[\large \frac{du}{dx}=-1\]We'll "multiply" the `dx` differential to the other side,\[\large du=-dx\]We'll multiply the `-1` to the other side,\[\large \color{orangered}{-du=dx}\]

We'll substitute the `orange` and `blue` pieces into this integral,\[\large 3\int\limits\limits\limits \sqrt{\color{royalblue}{6-x}}\; \color{orangered}{dx}\]

Answer 4

\[\large 3\int\limits\limits \sqrt{\color{royalblue}{6-x}}\; \color{orangered}{dx} \qquad \rightarrow \qquad 3\int\limits\limits \sqrt{\color{royalblue}{u}}\; (\color{orangered}{-du})\]

Answer 5

Pull the negative outside the integral, then rewrite your root as a `rational expression`.\[\large -3 \int\limits u^{1/2}du\]Then from here, just apply the `Power Rule for Integration`.

Answer 6

Once you get more comfortable with problems like this, you won't need to do a U substitution. You'll be able to do that step in your head.

But for now, it's helpful to learn it this way.
Let me know if you're confused about how to continue.