Question

evaluate the integral.

$$\int_{5}^{3}(2x + 5)dx$$

$$\int_{5}^{3}(2x + 5)dx = \square$$ (simplify your answer.)

Explanation:

Step1: Find the antiderivative

The antiderivative of \(2x\) is \(x^{2}\) (using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\), here \(n = 1\)), and the antiderivative of \(5\) is \(5x\). So the antiderivative of \(2x + 5\) is \(F(x)=x^{2}+5x\).

Step2: Apply the fundamental theorem of calculus

The fundamental theorem of calculus states that \(\int_{a}^{b}f(x)dx=F(b)-F(a)\). Here \(a = 5\), \(b = 3\), \(F(x)=x^{2}+5x\). Then \(F(3)-F(5)=(3^{2}+5\times3)-(5^{2}+5\times5)\).
First, calculate \(3^{2}+5\times3=9 + 15=24\).
Second, calculate \(5^{2}+5\times5=25+25 = 50\).
So \(F(3)-F(5)=24-50=-26\).

Answer:

\(-26\)