Problem 8 - Essential Calculus

Question

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{1}^{3}\left(3 x^{2}+5 x-4\right) d x $$

Step-by-Step Solution

Answer

The definite integral of the given function from 1 to 3 is 38

Step 1: Apply the integral separately to each term

The given function is a polynomial and can be separated into individual terms when integrating. Apply the integral to each term separately:\( \int_{1}^{3}(3x^{2}) dx + \int_{1}^{3}(5x) dx - \int_{1}^{3}(4) dx \)

Step 2: Evaluate the integrals

Use the power rule to evaluate the integrals. The power rule states that the integral of \(x^n dx\) is \(\frac{1}{n+1}x^{n+1}\).After applying power rule,\( [x^{3}]_{1}^{3} + [\frac{5}{2}x^{2}]_{1}^{3} - [4x]_{1}^{3} \)

Step 3: Substitute the limits

Substitute upper limit (3) first then subtract the result after substituting lower limit (1):\( (3^3 - 1^3) + \frac{5}{2}(3^2 - 1^2) - 4(3 - 1) \)

Step 4: Simplify to get the final result

Once all operations have been performed, the result will be obtained:\( (27 - 1) + \frac{5}{2}(9 - 1) - 4(2) = 26 + 20 - 8 = 38 \)