How to Evaluate the Definite Integral of a Sum

Step-by-step solution

1. Find the antiderivative term by term.

   $$\int (3x^2 + 2x)\,dx = x^3 + x^2$$

   Why: Integrate each term with the power rule.

2. Write the limits on the bracket.

   $$\left[x^3 + x^2\right]_0^2$$

   Why: The limits stay on the bracket until both ends are evaluated.

3. Evaluate at the top and bottom limits.

   $$(2^3 + 2^2) - (0^3 + 0^2) = 12 - 0$$

   Why: Substitute the top limit, then the bottom limit, and subtract.

4. Subtract.

   $$12 - 0 = 12$$

   Why: The result is the signed area under the graph.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $\int_0^1 (2x + 1)\,dx$

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   $2$

2. 2

   $\int_0^2 (x^2 + x)\,dx$

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   $\frac{14}{3}$

3. 3

   $\int_1^2 (3x^2 + 2x)\,dx$

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   $10$

4. 4

   $\int_0^1 (4x^3 + 2x)\,dx$

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   $2$

5. 5

   $\int_0^2 (6x^2 + 2)\,dx$

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   $20$