Anti-derivative of 4x / (x^2 + 6) u-Substitution

Suppose we need to solve the following integral...

Now, the normal inclination when we have an x² term on the bottom is to use a trigonometric substitution. For instance, by letting x = tanθ, as we've got an x² plus a positive number here.

But in this case, if we were to do that, it would actually be more work than what's necessary to
solve the integral, simply because we have this 4x term on the top.

Instead, we can rewrite this integral as...

Why? Well, 2x is the derivative of x², and that means it will cancel that when we use a simple u-substitution. So let u = x² + 6, and we will transform this from an integral with respect to x to an integral with respect to u.

So if we differentiate u with respect to x, we'll get du/dx = 2x. And that implies that dx = du/2x

Now, the the integral can be re-written as...

Please watch the video for the complete tutorial.