Solve ∫ (from 0 to 1) of xxy^2 wrt ydx

$$\int _ { 0 } ^ { 1 } x x y ^ { 2 } d y d x$$

Evaluate

$$\frac{dy^{3}}{3}$$

Solution Steps

Multiply x and x to get x^{2}.

$$\int _{0}^{1}x^{2}y^{2}dy\mathrm{d}x$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\int _{0}^{1}x^{2}y^{3}d\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int x^{2}y^{3}d\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$y^{3}d\int x^{2}\mathrm{d}x$$

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

$$y^{3}d\times \frac{x^{3}}{3}$$

Simplify.

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

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.

$$\frac{1}{3}y^{3}d\times 1^{3}-\frac{1}{3}y^{3}d\times 0^{3}$$

Simplify.

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

Differentiate w.r.t. y

$$dy^{2}$$