Sure! Let's carefully go through the given problem step-by-step.
### Problem Statement
We are given a function [tex]\( f(x) = x^4 - a x^3 + b x^2 + x - 5 \)[/tex] and we need to integrate this function from 0 to 2 and set the result to 24. We will then solve for the coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
### Given Function
[tex]\[ f(x) = x^4 - a x^3 + b x^2 + x - 5 \][/tex]
### Step 1: Integrate [tex]\( f(x) \)[/tex] from 0 to 2
To find the integral of [tex]\( f(x) \)[/tex] from 0 to 2, we need to compute:
[tex]\[ \int_{0}^{2} (x^4 - a x^3 + b x^2 + x - 5) \, dx \][/tex]
### Step 2: Compute the Indefinite Integral
Let's integrate each term of the polynomial separately:
1. [tex]\(\int x^4 \, dx = \frac{x^5}{5}\)[/tex]
2. [tex]\(\int -a x^3 \, dx = -a \frac{x^4}{4} = -\frac{a x^4}{4}\)[/tex]
3. [tex]\(\int b x^2 \, dx = b \frac{x^3}{3} = \frac{b x^3}{3}\)[/tex]
4. [tex]\(\int x \, dx = \frac{x^2}{2}\)[/tex]
5. [tex]\(\int -5 \, dx = -5x\)[/tex]
Therefore, the indefinite integral is:
[tex]\[
\int (x^4 - a x^3 + b x^2 + x - 5) \, dx = \frac{x^5}{5} - \frac{a x^4}{4} + \frac{b x^3}{3} + \frac{x^2}{2} - 5x + C
\][/tex]
### Step 3: Evaluate the Definite Integral
Now, we need to evaluate this indefinite integral from 0 to 2:
[tex]\[
\left[ \frac{x^5}{5} - \frac{a x^4}{4} + \frac{b x^3}{3} + \frac{x^2}{2} - 5x \right]_0^2
\][/tex]
Evaluate at the upper limit (x = 2):
[tex]\[
\left( \frac{2^5}{5} - \frac{a (2^4)}{4} + \frac{b (2^3)}{3} + \frac{2^2}{2} - 5 \cdot 2 \right)
\][/tex]
Simplify:
[tex]\[
\left( \frac{32}{5} - \frac{a (16)}{4} + \frac{b (8)}{3} + \frac{4}{2} - 10 \right)
\][/tex]
Then, evaluate at the lower limit (x = 0):
[tex]\[
\left( \frac{0}{5} - \frac{a (0)}{4} + \frac{b (0)}{3} + \frac{0}{2} - 5 \cdot 0 \right) = 0
\][/tex]
Subtracting upper limit evaluation from lower limit evaluation:
[tex]\[
\frac{32}{5} - 4a + \frac{8b}{3} + 2 - 10 - 0
\][/tex]
Simplify:
[tex]\[
\frac{32}{5} + 2 - 10 - 4a + \frac{8b}{3}
\][/tex]
Combine the constants:
[tex]\[
\frac{32}{5} + 2 - 10 = \frac{32}{5} + 2 - 10 = 6.4 + 2 - 10 = -1.6
\][/tex]
So, the equation becomes:
[tex]\[
-4a + \frac{8b}{3} - 1.6
\][/tex]
### Step 4: Set the Integral Equal to 24
We are given that the integral is equal to 24:
[tex]\[
-4a + \frac{8b}{3} - 1.6 = 24
\][/tex]
### Step 5: Solve the Equation
Rewriting the equation:
[tex]\[
-4a + \frac{8b}{3} - 1.6 = 24
\][/tex]
Add 1.6 to both sides:
[tex]\[
-4a + \frac{8b}{3} = 25.6
\][/tex]
Multiply through by 3 to clear the fraction:
[tex]\[
-12a + 8b = 76.8
\][/tex]
Divide through by 4:
[tex]\[
-3a + 2b = 19.2
\][/tex]
### Step 6: Solving for Variables
Using this linear equation, we solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
Below is one set of possible solutions:
[tex]\[
a = 0.666666666666667b - 6.4
\][/tex]
Given [tex]\(b\)[/tex], you can calculate [tex]\(a\)[/tex]. If [tex]\(b = 6\)[/tex]:
[tex]\[
a = 0.666666666666667(6) - 6.4 = 4 - 6.4 = -2.4
\][/tex]
So, a particular solution pair could be [tex]\( a = -2.4 \)[/tex] and [tex]\( b = 6 \)[/tex], which can be further refined with other values.
Therefore, the steps lead to solving for the unknown coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
