The problem asks to evaluate the integral $\int 2^x dx$. The given solution is $\frac{2^{x+1}}{x+1} + c$. We need to check if this is the correct answer.

1. Problem Description

The problem asks to evaluate the integral $\int 2^x dx$. The given solution is $\frac{2^{x+1}}{x+1} + c$. We need to check if this is the correct answer.

2. Solution Steps

To check if the proposed solution is correct, we can differentiate it and see if we get back the original integrand $2^x$.

Let $F(x) = \frac{2^{x+1}}{x+1} + c$.

We need to find $F'(x)$. We will use the quotient rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$. Here, $u = 2^{x+1}$ and $v = x+1$.

The derivative of $v=x+1$ is $v' = 1$.

The derivative of $u=2^{x+1}$ is $u' = \frac{d}{dx} 2^{x+1} = 2^{x+1} \ln(2) \cdot \frac{d}{dx}(x+1) = 2^{x+1} \ln(2)$.

Therefore,

$F'(x) = \frac{(2^{x+1} \ln(2))(x+1) - (2^{x+1})(1)}{(x+1)^2} = \frac{2^{x+1} ((x+1) \ln(2) - 1)}{(x+1)^2}$.

Since $F'(x) \ne 2^x$, the provided solution is incorrect.

The correct integral can be found using the formula:

$\int a^x dx = \frac{a^x}{\ln(a)} + C$

In this case, $a=2$, so

$\int 2^x dx = \frac{2^x}{\ln(2)} + C$.

3. Final Answer

The correct integral is $\frac{2^x}{\ln(2)} + C$.

The given solution is incorrect.

$\int 2^x dx = \frac{2^x}{\ln 2} + C$