Solve ∫ (from 10 to 30) of 2x^2+12y

$$\int _ { 10 } ^ { 30 } 2 x ^ { 2 } + 12 y$$

Evaluate

$$240y+\frac{52000}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2x^{2}+12y\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x^{2}\mathrm{d}x+\int 12y\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{2}\mathrm{d}x+12\int y\mathrm{d}x$$

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

$$\frac{2x^{3}}{3}+12\int y\mathrm{d}x$$

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

$$\frac{2x^{3}}{3}+12yx$$

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.

$$\frac{2}{3}\times 30^{3}+12y\times 30-\left(\frac{2}{3}\times 10^{3}+12y\times 10\right)$$

Simplify.

$$\frac{52000}{3}+240y$$

Differentiate w.r.t. y

$$240$$