Solve ∫ (from 5 to 6) of x^2+10x+25/x+5=

$$\int _ { 5 } ^ { 6 } \frac { x ^ { 2 } + 10 x + 25 } { x + 5 } =$$

Evaluate

$$\frac{21}{2}=10.5$$

Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}+10x+25}{x+5}.

$$\int _{5}^{6}\frac{\left(x+5\right)^{2}}{x+5}\mathrm{d}x$$

Cancel out x+5 in both numerator and denominator.

$$\int _{5}^{6}x+5\mathrm{d}x$$

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.

$$5\times 6+\frac{6^{2}}{2}-\left(5\times 5+\frac{5^{2}}{2}\right)$$

Simplify.

$$\frac{21}{2}$$