Solve ∫ (from 0 to x ^ 2) of xy wrt y

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

Evaluate

$$\frac{x^{5}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int xy\mathrm{d}y$$

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

$$x\int y\mathrm{d}y$$

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

$$x\times \frac{y^{2}}{2}$$

Simplify.

$$\frac{xy^{2}}{2}$$

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.

$$\frac{1}{2}x\left(x^{2}\right)^{2}-\frac{1}{2}x\times 0^{2}$$

Simplify.

$$\frac{x^{5}}{2}$$

Differentiate w.r.t. x

$$\frac{5x^{4}}{2}$$