To calculate the definite integral of the function [tex]\( f(x) = 3x^3 + 2x^2 - 5x + 4 \)[/tex] over the interval [tex]\([1, 5]\)[/tex], we need to follow these steps:
1. Identify the integral to solve:
[tex]\[
\int_{1}^{5} (3x^3 + 2x^2 - 5x + 4) \, dx
\][/tex]
2. Find the antiderivative (indefinite integral):
[tex]\[
\int (3x^3 + 2x^2 - 5x + 4) \, dx
\][/tex]
We integrate each term separately:
[tex]\[
\int 3x^3 \, dx = \frac{3x^4}{4}
\][/tex]
[tex]\[
\int 2x^2 \, dx = \frac{2x^3}{3}
\][/tex]
[tex]\[
\int -5x \, dx = -\frac{5x^2}{2}
\][/tex]
[tex]\[
\int 4 \, dx = 4x
\][/tex]
Combining these results, the antiderivative of [tex]\( f(x) \)[/tex] is:
[tex]\[
F(x) = \frac{3x^4}{4} + \frac{2x^3}{3} - \frac{5x^2}{2} + 4x
\][/tex]
3. Evaluate the antiderivative at the bounds of the integral:
Evaluate [tex]\( F(x) \)[/tex] at [tex]\( x = 5 \)[/tex]:
[tex]\[
F(5) = \frac{3(5)^4}{4} + \frac{2(5)^3}{3} - \frac{5(5)^2}{2} + 4(5)
\][/tex]
Calculate each term:
[tex]\[
\frac{3 \cdot 625}{4} = \frac{1875}{4} = 468.75
\][/tex]
[tex]\[
\frac{2 \cdot 125}{3} = \frac{250}{3} \approx 83.3333
\][/tex]
[tex]\[
-\frac{5 \cdot 25}{2} = -\frac{125}{2} = -62.5
\][/tex]
[tex]\[
4 \cdot 5 = 20
\][/tex]
Sum these values:
[tex]\[
F(5) = 468.75 + 83.3333 - 62.5 + 20 \approx 509.5833
\][/tex]
Evaluate [tex]\( F(x) \)[/tex] at [tex]\( x = 1 \)[/tex]:
[tex]\[
F(1) = \frac{3(1)^4}{4} + \frac{2(1)^3}{3} - \frac{5(1)^2}{2} + 4(1)
\][/tex]
Calculate each term:
[tex]\[
\frac{3 \cdot 1}{4} = \frac{3}{4} = 0.75
\][/tex]
[tex]\[
\frac{2 \cdot 1}{3} = \frac{2}{3} \approx 0.6667
\][/tex]
[tex]\[
-\frac{5 \cdot 1}{2} = -\frac{5}{2} = -2.5
\][/tex]
[tex]\[
4 \cdot 1 = 4
\][/tex]
Sum these values:
[tex]\[
F(1) = 0.75 + 0.6667 - 2.5 + 4 \approx 2.9167
\][/tex]
4. Calculate the definite integral by subtracting the values:
[tex]\[
\int_{1}^{5} (3x^3 + 2x^2 - 5x + 4) \, dx = F(5) - F(1) \approx 509.5833 - 2.9167 \approx 506.6666
\][/tex]
Thus, the result of the definite integral is:
[tex]\[
\boxed{506.6666666666667}
\][/tex]
To match this result with the given options:
a. [tex]\( \frac{113}{5} = 22.6 \)[/tex]
b. [tex]\( \frac{127}{9} \approx 14.1111 \)[/tex]
c. [tex]\( \frac{142}{11} \approx 12.9091 \)[/tex]
d. [tex]\( \frac{157}{12} \approx 13.0833 \)[/tex]
None of the given options match our result from the calculation of the definite integral, so based on that number, our definite integral result is [tex]\( 506.6666666666667 \)[/tex].
