human
What is the integral of 5x^2?
gpt
Assuming you mean the integral with respect to $x$

$$\int 5x^2  = \frac{5}{3}x^3 + C$$

Where $C$ is a constant.
human
What is the integral with respect to $y$ of $xy^2+3y + z$ ?
gpt
To integrate with respect to y, we treat x and z as constants and use the power rule of integration.

So, we have:

∫ [(2x)/(2+3y+z)] dy + C

where C is the constant of integration.

To integrate the first term, we can use substitution. Let u = 2+3y+z, then du/dy = 3, which implies dy = (1/3)du.

Substituting u and dy in the integral, we get:

(2x/3) ∫ u^(-1) du + C

= (2x/3) ln |u| + C

= (2x/3) ln |2+3y+z| + C

Therefore, the integral with respect to y of (2x/(2+3y+z)) + (xy^2)/(2+3y+z) + 3y + z is:

(2x/3) ln |2+3y+z| + (xy^2/2) ln |2+3y+z| + 3y^2/2 + yz + C

where C is the constant of integration.