Question

question 3
your mathematical work must accurately support your final answer. correct answers receive no credit without mathematically correct supporting work. do not approximate. no credit for decimal number representations.
(10 points) given \\( \frac { d y } { d x } = x - \sin x \\), find \\( y \\) where \\( y ( \pi ) = 3 \\).
click to proceed when you have completed the work on your own paper.

Explanation:

Step1: Integrate the derivative

Integrate \(\frac{dy}{dx}=x - \sin x\) with respect to \(x\).
Using the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)) and \(\int\sin xdx=-\cos x + C\), we have:
\(y=\int(x-\sin x)dx=\int xdx-\int\sin xdx=\frac{x^{2}}{2}+\cos x + C\)

Step2: Substitute the initial condition

Given \(y(\pi)=3\), substitute \(x = \pi\) and \(y = 3\) into \(y=\frac{x^{2}}{2}+\cos x + C\).
\(3=\frac{\pi^{2}}{2}+\cos(\pi)+C\)
Since \(\cos(\pi)=- 1\), we get \(3=\frac{\pi^{2}}{2}-1 + C\)
Solve for \(C\): \(C=4-\frac{\pi^{2}}{2}\)

Answer:

\(y=\frac{x^{2}}{2}+\cos x+4-\frac{\pi^{2}}{2}\)