Solve ∫ (from 0 to 2) of x^2-2x-15/(x-5) wrt x

$$\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } - 2 x - 15 } { ( x - 5 ) } d x$$

Evaluate

$$8$$

Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-2x-15}{x-5}.

$$\int _{0}^{2}\frac{\left(x-5\right)\left(x+3\right)}{x-5}\mathrm{d}x$$

Cancel out x-5 in both numerator and denominator.

$$\int _{0}^{2}x+3\mathrm{d}x$$

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

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

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

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

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.

$$3\times 2+\frac{2^{2}}{2}-\left(3\times 0+\frac{0^{2}}{2}\right)$$

Simplify.

$$8$$