QuestionFind the general antiderivative of $\frac{d y}{d x}=7 e^{x}+4$ . Antiderivative $=$ $+C$ .

Studdy Solution

STEP 1

Assumptions1. The function given is $\frac{d y}{d x}=7 e^{x}+4$ . We are asked to find the most general antiderivative of this function3. We are not to include the constant of integration $+C$ in our steps, as it is understood to be included in the final answer

STEP 2

The antiderivative of a function is the function whose derivative is the given function. In this case, we need to find a function whose derivative is $7 e^{x}+4$ .

STEP 3

We can find the antiderivative by integrating the given function. The integral of a sum of functions is equal to the sum of the integrals of the functions. So, we can write $\int \frac{d y}{d x} dx = \int (7 e^{x}+) dx = \int7 e^{x} dx + \int dx$

STEP 4

We can now integrate each term separately. The integral of $e^{x}$ with respect to $x$ is $e^{x}$ , and the integral of a constant is the constant times $x$ . So, we have $\int7 e^{x} dx + \int4 dx =7 \int e^{x} dx +4 \int dx =7 e^{x} +4x$ The most general antiderivative of the function $\frac{d y}{d x}=7 e^{x}+4$ is $7 e^{x} +4x + C$ , where $C$ is the constant of integration.

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