`int(x2^(x))dx=?`

Options

A. `(2^(x))/((log2))(x+log2)+C`
B. `(2^(x))/((log2))(xlog2-1)+C`
C. `(x*2^(x))/((log2))+(2^(x))/((log2)^(2))+C`
D. none of these

Explanation

To solve the integral \( \int x \cdot 2^x \, dx \), we will use the method of integration by parts. The formula for integration by parts is given by:

\[ \int u \, dv = uv - \int v \, du \]

### Step 1: Choose \( u \) and \( dv \)

Let:
- \( u = x \) (first function)
- \( dv = 2^x \, dx \) (second function)

### Step 2: Differentiate \( u \) and integrate \( dv \)

Now, we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = dx \]

- Integrate \( dv \):
\[ v = \int 2^x \, dx = \frac{2^x}{\ln(2)} \]

### Step 3: Apply the integration by parts formula

Substituting into the integration by parts formula:
\[ \int x \cdot 2^x \, dx = uv - \int v \, du \]

This gives us:
\[ \int x \cdot 2^x \, dx = x \cdot \frac{2^x}{\ln(2)} - \int \frac{2^x}{\ln(2)} \, dx \]

### Step 4: Simplify the integral

Now we need to evaluate the integral \( \int \frac{2^x}{\ln(2)} \, dx \):
\[ \int \frac{2^x}{\ln(2)} \, dx = \frac{1}{\ln(2)} \int 2^x \, dx = \frac{1}{\ln(2)} \cdot \frac{2^x}{\ln(2)} = \frac{2^x}{\ln(2)^2} \]

### Step 5: Substitute back into the equation

Now substitute this result back into our equation:
\[ \int x \cdot 2^x \, dx = x \cdot \frac{2^x}{\ln(2)} - \frac{2^x}{\ln(2)^2} \]

### Step 6: Factor out common terms

Factoring out \( \frac{2^x}{\ln(2)^2} \):
\[ \int x \cdot 2^x \, dx = \frac{2^x}{\ln(2)^2} \left( x \ln(2) - 1 \right) + C \]

### Final Answer

Thus, the final result is:
\[ \int x \cdot 2^x \, dx = \frac{2^x}{\ln(2)^2} \left( x \ln(2) - 1 \right) + C \]