Evaluate a definite integral using FTC Part 2

Evaluate the definite integral $\int_1^3 (x^2 + 2x)\, dx$ using the Fundamental Theorem of Calculus, Part 2. What is the exact value of this integral?

Hint

First find an antiderivative $F(x)$ of $f(x) = x^2 + 2x$ . Then compute $F(3) - F(1)$ using the evaluation bar notation $F(x)\Big|_1^3$ .

Step-by-step solution

An antiderivative of $f(x) = x^2 + 2x$ is $F(x) = \dfrac{x^3}{3} + x^2$ , since $F'(x) = x^2 + 2x$ . By FTC Part 2, $\int_1^3 (x^2+2x)\,dx = F(x)\Big|_1^3 = F(3) - F(1)$ .

Compute $F(3) = \dfrac{27}{3} + 9 = 9 + 9 = 18$ , and $F(1) = \dfrac{1}{3} + 1 = \dfrac{4}{3}$ . Subtracting gives $18 - \dfrac{4}{3} = \dfrac{54}{3} - \dfrac{4}{3} = \dfrac{50}{3}$ .

Identifying the values subtracted in FTC Part 2

Hint

Remember that by FTC Part 2, you subtract the antiderivative evaluated at the lower limit from the antiderivative evaluated at the upper limit.

Step-by-step solution

Evaluate a definite integral using FTC Part 2

Identifying the values subtracted in FTC Part 2

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