Question

integrate.\\(int left( \frac{2}{x} - 4e^x
ight) dx = ?\\)\\(\text{choose 1 answer:}\\)\\(\text{a } 2ln|x| - e^{4x} + c\\)\\(\text{b } 2ln(x) - 4e^x + c\\)\\(\text{c } 2ln(x) - e^{4x} + c\\)\\(\text{d } 2ln|x| - 4e^x + c\\)

Explanation:

Step1: Integrate \(\frac{2}{x}\)

The integral of \(\frac{1}{x}\) is \(\ln|x|\), so the integral of \(\frac{2}{x}\) is \(2\ln|x|\) (using the constant multiple rule: \(\int kf(x)dx = k\int f(x)dx\) where \(k = 2\) and \(f(x)=\frac{1}{x}\)).

Step2: Integrate \(-4e^{x}\)

The integral of \(e^{x}\) is \(e^{x}\), so the integral of \(-4e^{x}\) is \(-4e^{x}\) (using the constant multiple rule: \(\int kf(x)dx = k\int f(x)dx\) where \(k=-4\) and \(f(x)=e^{x}\)).

Step3: Add the constant of integration \(C\)

When integrating, we must include the constant of integration \(C\) to account for all possible antiderivatives. Combining the results from Step 1 and Step 2, we get \(2\ln|x|-4e^{x}+C\).

Answer:

D. \(2\ln|x| - 4e^{x}+C\)