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QuestionFind an antiderivative of f(x)=2x−10exf(x)=\frac{2}{x}-10 e^{x}.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=x−10exf(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)f(x).
∫f(x) dx=∫(2x−10ex) dx\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.
∫(2x−10ex) dx=∫2x dx−∫10ex dx\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 1x\frac{1}{x} is ln∣x∣ln|x|, and the integral of exe^{x} is exe^{x}.∫2x dx−∫10ex dx=2∫1x dx−10∫ex dx\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∫1x dx−10∫ex dx=2ln⁡∣x∣−10ex2 \int \frac{1}{x} \, dx -10 \int e^{x} \, dx =2 \ln|x| -10 e^{x}

STEP 6

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.
2ln⁡∣x∣−10ex+C2 \ln|x| -10 e^{x} + CSo, an antiderivative of the function f(x)=2x−10exf(x)=\frac{2}{x}-10 e^{x} is (x)=2ln⁡∣x∣−10ex+C(x) =2 \ln|x| -10 e^{x} + C.

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