Solve ∫ (from 0 to a) of (y-2x)

$$\int _ { 0 } ^ { a } ( y - 2 x )$$

Evaluate

$$a\left(y-a\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int y-2x\mathrm{d}x$$

Integrate the sum term by term.

$$\int y\mathrm{d}x+\int -2x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int y\mathrm{d}x-2\int x\mathrm{d}x$$

Find the integral of y using the table of common integrals rule \int a\mathrm{d}x=ax.

$$yx-2\int x\mathrm{d}x$$

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}. Multiply -2 times \frac{x^{2}}{2}.

$$yx-x^{2}$$

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.

$$ya-a^{2}-\left(y\times 0-0^{2}\right)$$

Simplify.

$$a\left(y-a\right)$$

Differentiate w.r.t. y

$$a$$