Solve ∫ (from 0 to frac{ 1) of {2}}5xy wrt y

$$\int_{ 0 }^{ \frac{ 1 }{ 2 } } 5xy d y$$

Evaluate

$$\frac{5x}{8}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 5xy\mathrm{d}y$$

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

$$5x\int y\mathrm{d}y$$

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}.

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

Simplify.

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

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.

$$\frac{5}{2}x\times \left(\frac{1}{2}\right)^{2}-\frac{5}{2}x\times 0^{2}$$

Simplify.

$$\frac{5x}{8}$$

Differentiate w.r.t. x

$$\frac{5}{8} = 0.625$$