Question
\( \int ( 4 x ^ { 11 } - 5 x ^ { 14 } ) d x = \square \)
Ask by Dickson Chambers. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\int \left( 4x^{11} - 5x^{14} \right) dx = \frac{x^{12}}{3} - \frac{x^{15}}{3} + C
\]
Solution
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:**
\[
\int \left( 4x^{11} - 5x^{14} \right) dx = \frac{x^{12}}{3} - \frac{x^{15}}{3} + C
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To solve the integral \( \int ( 4 x^{11} - 5 x^{14} ) d x \), we can integrate each term separately. For the first term, \( \int 4 x^{11} d x = 4 \cdot \frac{x^{12}}{12} = \frac{1}{3} x^{12} \). For the second term, \( \int -5 x^{14} d x = -5 \cdot \frac{x^{15}}{15} = -\frac{1}{3} x^{15} \). Combining these results, we have: \[ \int ( 4 x^{11} - 5 x^{14} ) d x = \frac{1}{3} x^{12} - \frac{1}{3} x^{15} + C \] where \( C \) is the constant of integration.
