We need to evaluate the definite integral of three x squared plus two x plus five from one to five. This represents the area under the curve between x equals one and x equals five.
To solve this definite integral, we first find the antiderivative. We integrate each term separately using the power rule. Three x squared becomes x cubed, two x becomes x squared, and five becomes five x. So our antiderivative F of x equals x cubed plus x squared plus five x.
Now we apply the Fundamental Theorem of Calculus. The definite integral from a to b equals F of b minus F of a. We calculate F of 5 equals 125 plus 25 plus 25 which equals 175. Then F of 1 equals 1 plus 1 plus 5 which equals 7. Finally, 175 minus 7 equals 168.
Here's our visual verification. The definite integral equals 168, which represents the exact area under the curve f of x equals three x squared plus two x plus five from x equals 1 to x equals 5. The red dashed lines show our integration boundaries.
To summarize our solution: We found the antiderivative F of x equals x cubed plus x squared plus five x. We applied the Fundamental Theorem of Calculus to get F of 5 minus F of 1. After calculating, we found F of 5 equals 175 and F of 1 equals 7, giving us the final answer of 168.