Question

find the most general antiderivative or indefinite integral.\\(\int \left(e^{-5x} + 2^{x}\
ight) dx\\)\\(\int \left(e^{-5x} + 2^{x}\
ight) dx = \square\\)

Explanation:

Step1: Integrate \( e^{-5x} \)

The integral of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \). For \( e^{-5x} \), \( a = -5 \), so \( \int e^{-5x}dx = \frac{1}{-5}e^{-5x} + C_1 = -\frac{1}{5}e^{-5x} + C_1 \).

Step2: Integrate \( 2^x \)

The integral of \( a^x \) is \( \frac{a^x}{\ln a} + C \). For \( 2^x \), \( a = 2 \), so \( \int 2^xdx = \frac{2^x}{\ln 2} + C_2 \).

Step3: Combine results

Using the sum rule of integration \( \int (f(x) + g(x))dx = \int f(x)dx + \int g(x)dx \), we add the two integrals and combine the constants \( C_1 \) and \( C_2 \) into a single constant \( C \). So \( \int (e^{-5x} + 2^x)dx = -\frac{1}{5}e^{-5x} + \frac{2^x}{\ln 2} + C \).

Answer:

\( -\frac{1}{5}e^{-5x} + \frac{2^x}{\ln 2} + C \)