Questions: Find the antiderivative for each function when C equals 0. a. f(x) = e^(10x) b. g(x) = e^(-5x) c. h(x) = e^(x/12)

Transcript text: Pearson MyLab and Mastering Course Home earson.com/Student/IntegratedAssignmentOverview.aspx?homeworkld=680005116 Course tive Assignment Question Part 1 of 3 Find the antiderivative for each function when C equals 0 . a. $f(x)=e^{10 x}$ b. $g(x)=e^{-5 x}$ c. $h(x)=e^{\frac{x}{12}}$

Solution

Solution Steps

To find the antiderivative of each function, we need to integrate the given exponential functions. The antiderivative of an exponential function of the form $ e^{ax} $ is $ \frac{1}{a} e^{ax} + C $. Since $ C = 0 $, we will not include the constant of integration in our final answer.

Step 1: Antiderivative of $ f(x) = e^{10x} $

To find the antiderivative of $ f(x) $, we integrate: \[ \int e^{10x} \, dx = \frac{1}{10} e^{10x} \] Since $ C = 0 $, the antiderivative is: \[ \frac{1}{10} e^{10x} \]

Step 2: Antiderivative of $ g(x) = e^{-5x} $

Next, we find the antiderivative of $ g(x) $: \[ \int e^{-5x} \, dx = -\frac{1}{5} e^{-5x} \] Again, with $ C = 0 $, the antiderivative is: \[ -\frac{1}{5} e^{-5x} \]

Step 3: Antiderivative of $ h(x) = e^{\frac{x}{12}} $

Finally, we compute the antiderivative of $ h(x) $: \[ \int e^{\frac{x}{12}} \, dx = 12 e^{\frac{x}{12}} \] With $ C = 0 $, the antiderivative is: \[ 12 e^{\frac{x}{12}} \]