Solve ∫ (from -2 to 4) of y+2x-x^2 wrt x

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

Evaluate

$$6\left(y-2\right)$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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}.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -1 times \frac{x^{3}}{3}.

$$yx+x^{2}-\frac{x^{3}}{3}$$

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.

$$y\times 4+4^{2}-\frac{4^{3}}{3}-\left(y\left(-2\right)+\left(-2\right)^{2}-\frac{\left(-2\right)^{3}}{3}\right)$$

Simplify.

$$-12+6y$$

Differentiate w.r.t. y

$$6$$