Question

homework9: problem 4
(1 point)
find the most general antiderivative for the function \\( \frac{d y}{d x}=8 e^{x}+5 \\).
note: dont enter the \\( +c \\). its included for you.
antiderivative \\( =\square+c \\)

Explanation:

Step1: Recall antiderivative rules

The antiderivative of \( e^x \) is \( e^x \) (since \( \frac{d}{dx}(e^x) = e^x \)) and the antiderivative of a constant \( k \) is \( kx \) (since \( \frac{d}{dx}(kx) = k \)). For the function \( \frac{dy}{dx}=8e^x + 5 \), we find the antiderivative term - by - term.

Step2: Find antiderivative of \( 8e^x \)

Using the rule that the antiderivative of \( e^x \) is \( e^x \), when we have \( 8e^x \), the antiderivative is \( 8e^x \) (because the derivative of \( 8e^x \) with respect to \( x \) is \( 8e^x \), by the constant multiple rule of differentiation: \( \frac{d}{dx}(cf(x))=c\frac{d}{dx}(f(x)) \), here \( c = 8 \) and \( f(x)=e^x \)).

Step3: Find antiderivative of \( 5 \)

Using the rule that the antiderivative of a constant \( k \) is \( kx \), for \( k = 5 \), the antiderivative is \( 5x \) (since \( \frac{d}{dx}(5x)=5 \)).

Step4: Combine the antiderivatives

Adding the antiderivatives of the two terms together, the antiderivative of \( 8e^x+5 \) is \( 8e^x + 5x \).

Answer:

\( 8e^x + 5x \)