$$ \int ( 4 x ^ { 11 } - 5 x ^ { 14 } ) d x = \square $$

Question

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To evaluate the integral:

$$
\int \left( 4x^{11} - 5x^{14} \right) dx
$$

We can integrate each term separately using the power rule for integration, which states that:

$$
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
$$

where $$ C $$ is the constant of integration.

1. **Integrate $$ 4x^{11} $$:**

$$
\int 4x^{11} \, dx = 4 \cdot \frac{x^{12}}{12} = \frac{x^{12}}{3}
$$

2. **Integrate $$ -5x^{14} $$:**

$$
\int -5x^{14} \, dx = -5 \cdot \frac{x^{15}}{15} = -\frac{x^{15}}{3}
$$

3. **Combine the results and add the constant of integration $$ C $$:**

$$
\int \left( 4x^{11} - 5x^{14} \right) dx = \frac{x^{12}}{3} - \frac{x^{15}}{3} + C
$$

**Final Answer:**

$$
\int \left( 4x^{11} - 5x^{14} \right) dx = \frac{x^{12}}{3} - \frac{x^{15}}{3} + C
$$
$$ \int \left( 4x^{11} - 5x^{14} \right) dx = \frac{x^{12}}{3} - \frac{x^{15}}{3} + C $$

\( \int ( 4 x ^ { 11 } - 5 x ^ { 14 } ) d x = \square \)

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