Solve ∫ (from x to x ^ 2) of y^2 wrt y

$$\int _ { x } ^ { x ^ { 2 } } y ^ { 2 } d y$$

Evaluate

$$\frac{x^{3}\left(x^{3}-1\right)}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int y^{2}\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{2}\mathrm{d}y with \frac{y^{3}}{3}.

$$\frac{y^{3}}{3}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{1}{3}\left(x^{2}\right)^{3}-\frac{x^{3}}{3}$$

Simplify.

$$\frac{-x^{3}+x^{6}}{3}$$

Differentiate w.r.t. x

$$x^{2}\left(2x^{3}-1\right)$$