Solve ∫ (from x to 2x) of 2x+4y wrt y

$$\int_{ x }^{ 2x } 2x+4y d y$$

Evaluate

$$8x^{2}$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

$$2xy+4\int y\mathrm{d}y$$

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}. Multiply 4 times \frac{y^{2}}{2}.

$$2xy+2y^{2}$$

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.

$$2x\times 2x+2\times \left(2x\right)^{2}-\left(2xx+2x^{2}\right)$$

Simplify.

$$8x^{2}$$

Differentiate w.r.t. x

$$16x$$