Solve ∫ (from -y to y) of xy wrt x

$$\int_{ -y }^{ y } xy d x$$

Evaluate

$$0$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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

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

Simplify.

$$\frac{yx^{2}}{2}$$

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.

$$\frac{1}{2}yy^{2}-\frac{1}{2}y\left(-y\right)^{2}$$

Simplify.

$$0$$

Differentiate w.r.t. y

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