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Sagot :
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].
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].
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