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int_(1)f(x)dx=

F(x)=|4x-20|

f(x)=F^(')(x)

$\int_{1} f(x) d x=$ $\newline$ $F(x)=|4 x-20|$ $\newline$ $f(x)=F^{\prime}(x)$

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Evaluate a nonlinear function

Full solution

Q.

\int_{1} f(x) d x=

\newline

F(x)=|4 x-20|

\newline

f(x)=F^{\prime}(x)

Find Derivative of F(x): First, we need to find the derivative of F(x) to get f(x). $\newline$ F(x) = |4x - 20|\(\newline\)To find $f(x)$ , we need to consider the absolute value function.
Consider Absolute Value Function: For $x > 5$ , $F(x) = 4x - 20$ , so $f(x) = F'(x) = 4$ . For $x < 5$ , $F(x) = -(4x - 20)$ , so $f(x) = F'(x) = -4$ .
Integrate $f(x)$ from $1$ to $x$ : Now, we integrate $f(x)$ from $1$ to $x$ . If $x > 5$ , the integral from $1$ to $x$ of $f(x)$ dx is the integral from $1$ to $1$ $1$ of $1$ $2$ dx plus the integral from $1$ $1$ to $x$ of $1$ $5$ dx.
Calculate Integral from $1$ to $5$ : Calculate the first part, integral from $1$ to $5$ of $-4$ dx. This is $-4$ times the integral from $1$ to $5$ of dx, which is $-4(x)|$ from $1$ to $5$ .
Calculate Integral from $5$ to $x$ : Evaluating from $1$ to $5$ gives us $-4(5 - 1) = -4(4) = -16$ .
Add Two Parts for Final Answer: Now, calculate the second part, integral from $5$ to $x$ of $4\,dx$ , if $x > 5$ . This is $4$ times the integral from $5$ to $x$ of $dx$ , which is $4(x)|$ from $5$ to $x$ .
Calculate Integral from $1$ to $x$ : Evaluating from $5$ to $x$ gives us $4(x - 5)$ .
Calculate Integral from $1$ to $x$ : Add the two parts together for the final answer if $x > 5$ . $\newline$ $-16 + 4(x - 5) = 4x - 36$ .
Final Answer: If $x < 5$ , the integral from $1$ to $x$ of $f(x)$ dx is just the integral from $1$ to $x$ of $-4$ dx.
Final Answer: If $x < 5$ , the integral from $1$ to $x$ of $f(x)dx$ is just the integral from $1$ to $x$ of $-4dx$ .Calculate the integral from $1$ to $x$ of $-4dx$ , which is $1$ $0$ from $1$ to $x$ .
Final Answer: If $x < 5$ , the integral from $1$ to $x$ of $f(x)dx$ is just the integral from $1$ to $x$ of $-4dx$ .Calculate the integral from $1$ to $x$ of $-4dx$ , which is $1$ $0$ from $1$ to $x$ .Evaluating from $1$ to $x$ gives us $1$ $5$ .
Final Answer: If $x < 5$ , the integral from $1$ to $x$ of $f(x)dx$ is just the integral from $1$ to $x$ of $-4dx$ . Calculate the integral from $1$ to $x$ of $-4dx$ , which is $1$ $0$ from $1$ to $x$ . Evaluating from $1$ to $x$ gives us $1$ $5$ . So, the final answer is $1$ $6$ if $1$ $7$ , and $1$ $5$ if $x < 5$ .

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