Solve ∫ (from c to n) of x+5 wrt x

$$\int _ { c } ^ { n } x + 5 d x$$

Evaluate

$$\frac{\left(n-c\right)\left(n+c+10\right)}{2}$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

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

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

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

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.

$$\frac{n^{2}}{2}+5n-\left(\frac{c^{2}}{2}+5c\right)$$

Simplify.

$$\frac{\left(10+c+n\right)\left(-c+n\right)}{2}$$

Differentiate w.r.t. c

$$-c-5$$