Substitution of a definite integral

[bobsmith76]

Homework Statement

Why does the 3x^2 disappear? Doesn't it play a role in the answer?

 

[jbunniii]

It is explained in the first line of blue text:

$$du = 3x^2 dx$$

 

[bobsmith76]

I think after you find the derivative of u, which is 3x^2, you divide that by 3x^2 which is 1, but I'm not sure

 

[jbunniii]

No, it's a simple substitution.

$$u = x^3 + 1$$

Differentiating both sides with respect to x:

$$\frac{du}{dx} = 3x^2$$

or equivalently

$$du = 3x^2 dx$$

Now substitute into the original integral. The $3x^2 dx$ turns into $du$, and $\sqrt{x^3 + 1}$ becomes $\sqrt{u}$. The endpoints of the integral also change accordingly.