Solve ∫ (from - 1 to 4) of 2x+5 wrt x

$$\int _ { - 1 } ^ { 4 } 2 x + 5 d x$$

Evaluate

$$40$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2x+5\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x\mathrm{d}x+\int 5\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x\mathrm{d}x+\int 5\mathrm{d}x$$

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

$$x^{2}+\int 5\mathrm{d}x$$

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

$$x^{2}+5x$$

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.

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

Simplify.

$$40$$