Question
\( \int _{}^{}(4x^{3}-6) d \)
Ask by Murphy Vaughan. in Ecuador
Jan 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
La integral de \(4x^{3} - 6\) con respecto a \(x\) es \(x^{4} - 6x + C\), donde \(C\) es una constante.
Solution
Para calcular la integral indefinida de \(4x^{3} - 6\) con respecto a \(x\), seguimos estos pasos:
\[
\int (4x^{3} - 6) \, dx
\]
1. **Integrar cada término por separado:**
\[
\int 4x^{3} \, dx - \int 6 \, dx
\]
2. **Aplicar las reglas de integración:**
\[
4 \cdot \frac{x^{4}}{4} - 6x + C
\]
3. **Simplificar la expresión:**
\[
x^{4} - 6x + C
\]
**Resultado Final:**
\[
\int (4x^{3} - 6) \, dx = x^{4} - 6x + C
\]
Donde \(C\) es la constante de integración.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Beyond the Answer
The integral of \( 4x^3 - 6 \) is actually quite straightforward! You can apply the power rule of integration. For the first term, \( 4x^3 \), you increase the exponent by one to get \( x^4 \) and divide by the new exponent, resulting in \( \frac{4}{4}x^4 = x^4 \). For the constant \(-6\), its integral is simply \(-6x\). So, the integral is \( x^4 - 6x + C \), where \( C \) is the constant of integration. Now, if you want to check your work, you can differentiate your result. The derivative of \( x^4 - 6x + C \) should bring you right back to \( 4x^3 - 6 \). If it does, you’ve nailed the integration!
