Evaluate the Integral. Check your answer by differentiating. (Remember the constant of integration. $$ \left(2^{2}+x^{11}\right) d x $$

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We need to evaluate

$$
\int \left(2^2 + x^{11}\right) \, dx.
$$

Since $$2^2 = 4$$, the integral becomes

$$
\int \left(4 + x^{11}\right) \, dx.
$$

### Step 1. Split the Integral

$$
\int \left(4 + x^{11}\right) \, dx = \int 4 \, dx + \int x^{11} \, dx.
$$

### Step 2. Evaluate Each Integral

1. For $$\int 4 \, dx$$:

$$
\int 4 \, dx = 4x.
$$

2. For $$\int x^{11} \, dx$$:

$$
\int x^{11} \, dx = \frac{x^{12}}{12}.
$$

### Step 3. Combine the Results and Add the Constant of Integration

$$
\int \left(4 + x^{11}\right) \, dx = 4x + \frac{x^{12}}{12} + C.
$$

### Step 4. Check the Result by Differentiating

Differentiate

$$
y = 4x + \frac{x^{12}}{12} + C
$$

with respect to $$x$$:

1. The derivative of $$4x$$ is $$4$$.
2. The derivative of $$\frac{x^{12}}{12}$$ is $$\frac{12x^{11}}{12} = x^{11}$$.
3. The derivative of $$C$$ is $$0$$.

Thus,

$$
y' = 4 + x^{11},
$$

which is the original integrand.

### Final Answer

$$
\boxed{4x + \frac{x^{12}}{12} + C.}
$$
The integral of $$2^2 + x^{11}$$ with respect to $$x$$ is $$4x + \frac{x^{12}}{12} + C$$.

Evaluate the Integral. Check your answer by differentiating. (Remember the constant of integration.

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