Solve ∫ (from 0 to 12) of (-13/90x^2+13/13x+72/5) wrt x

$$\int _ { 0 } ^ { 12 } ( - \frac { 13 } { 90 } x ^ { 2 } + \frac { 13 } { 13 } x + \frac { 72 } { 5 } ) d x$$

Evaluate

$$\frac{808}{5}=161.6$$

Solution Steps

Divide 13 by 13 to get 1.

$$\int _{0}^{12}-\frac{13}{90}x^{2}+1x+\frac{72}{5}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -\frac{13x^{2}}{90}+x+\frac{72}{5}\mathrm{d}x$$

Integrate the sum term by term.

$$\int -\frac{13x^{2}}{90}\mathrm{d}x+\int x\mathrm{d}x+\int \frac{72}{5}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-\frac{13\int x^{2}\mathrm{d}x}{90}+\int x\mathrm{d}x+\int \frac{72}{5}\mathrm{d}x$$

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 -\frac{13}{90} times \frac{x^{3}}{3}.

$$-\frac{13x^{3}}{270}+\int x\mathrm{d}x+\int \frac{72}{5}\mathrm{d}x$$

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}.

$$-\frac{13x^{3}}{270}+\frac{x^{2}}{2}+\int \frac{72}{5}\mathrm{d}x$$

Find the integral of \frac{72}{5} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{13x^{3}}{270}+\frac{x^{2}}{2}+\frac{72x}{5}$$

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.

$$-\frac{13}{270}\times 12^{3}+\frac{12^{2}}{2}+\frac{72}{5}\times 12-\left(-\frac{13}{270}\times 0^{3}+\frac{0^{2}}{2}+\frac{72}{5}\times 0\right)$$

Simplify.

$$\frac{808}{5}$$