Math Snap

STEP 1

Assumptions1. The function is $f(x)=\frac{}{x}-10 e^{x}$
. We are looking for an antiderivative of the function, which means we want to find a function F(x) such that F'(x) = f(x)

STEP 2

The antiderivative of a function is found by integrating the function. So we need to integrate $f(x)$.
$$\int f(x) \, dx = \int \left(\frac{2}{x}-10 e^{x}\right) \, dx$$

STEP 3

We can split the integral into two separate integrals because of the subtraction operation in the function.
$$\int \left(\frac{2}{x}-10 e^{x}\right) \, dx = \int \frac{2}{x} \, dx - \int10 e^{x} \, dx$$

STEP 4

Now, we can integrate each part separately. The integral of $\frac{1}{x}$ is $ln|x|$, and the integral of $e^{x}$ is $e^{x}$.$$\int \frac{2}{x} \, dx - \int10 e^{x} \, dx =2 \int \frac{1}{x} \, dx -10 \int e^{x} \, dx$$

STEP 5

Now, integrate each part.
$$2 \int \frac{1}{x} \, dx -10 \int e^{x} \, dx =2 \ln|x| -10 e^{x}$$

Finally, we add the constant of integration, C, to the result. This is because the antiderivative is not unique and can differ by a constant.
$$2 \ln|x| -10 e^{x} + C$$So, an antiderivative of the function $f(x)=\frac{2}{x}-10 e^{x}$ is $(x) =2 \ln|x| -10 e^{x} + C$.