Solve ∫ (from -1 to 5) of 1/2x+3 wrt x

$$\int_{ -1 }^{ 5 } \frac{ 1 }{ 2 } x+3 d x$$

Evaluate

$$24$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 \frac{1}{2} times \frac{x^{2}}{2}.

$$\frac{x^{2}}{4}+\int 3\mathrm{d}x$$

Find the integral of 3 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{2}}{4}+3x$$

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.

$$\frac{5^{2}}{4}+3\times 5-\left(\frac{\left(-1\right)^{2}}{4}+3\left(-1\right)\right)$$

Simplify.

$$24$$