Solve ∫ (from 3 to 12) of 7x+17117/18 wrt x

$$\int_{ 3 }^{ 12 } 7x+171 \frac{ 17 }{ 18 } d x$$

Evaluate

$$2020$$

Solution Steps

Multiply 171 and 18 to get 3078.

$$\int _{3}^{12}7x+\frac{3078+17}{18}\mathrm{d}x$$

Add 3078 and 17 to get 3095.

$$\int _{3}^{12}7x+\frac{3095}{18}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 7x+\frac{3095}{18}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 7x\mathrm{d}x+\int \frac{3095}{18}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$7\int x\mathrm{d}x+\int \frac{3095}{18}\mathrm{d}x$$

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 7 times \frac{x^{2}}{2}.

$$\frac{7x^{2}}{2}+\int \frac{3095}{18}\mathrm{d}x$$

Find the integral of \frac{3095}{18} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{7x^{2}}{2}+\frac{3095x}{18}$$

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.

$$\frac{7}{2}\times 12^{2}+\frac{3095}{18}\times 12-\left(\frac{7}{2}\times 3^{2}+\frac{3095}{18}\times 3\right)$$

Simplify.

$$2020$$