Solve ∫ (from 2 to 2 pi) of 2x-4y=

$$\int _ { 2 } ^ { 2 \pi } 2 x - 4 y =$$

Evaluate

$$4\left(-2\pi y+2y+\pi ^{2}-1\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2x-4y\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x\mathrm{d}x+\int -4y\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x\mathrm{d}x-4\int y\mathrm{d}x$$

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}.

$$x^{2}-4\int y\mathrm{d}x$$

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

$$x^{2}-4yx$$

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.

$$\left(2\pi \right)^{2}-4y\times 2\pi -\left(2^{2}-4y\times 2\right)$$

Simplify.

$$4\pi ^{2}-8y\pi -4+8y$$

Differentiate w.r.t. y

$$8-8\pi$$