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Answer
The integral of \(5x^2\) with respect to \(x\) is \(\frac{5x^3}{3} + C\).
Solution
Calculate or simplify the expression \( \int 5x^2 dx \).
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
\(\int 5x^{2} dx\)
- step1: Use properties of integrals:
\(5\times \int x^{2} dx\)
- step2: Evaluate the integral:
\(5\times \frac{x^{2+1}}{2+1}\)
- step3: Simplify:
\(5\times \frac{x^{3}}{3}\)
- step4: Multiply the terms:
\(\frac{5x^{3}}{3}\)
- step5: Add the constant of integral C:
\(\frac{5x^{3}}{3} + C, C \in \mathbb{R}\)
The integral of \(5x^2\) with respect to \(x\) is \(\frac{5x^3}{3} + C\), where \(C\) is a constant.
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Simplify this solution Beyond the Answer
To solve the integral \( \int 5 x^{2} d x \), you can apply the power rule for integration. When you integrate \( x^n \), you increase the exponent by one and divide by the new exponent. So, here you get \( 5 \times \frac{x^{3}}{3} + C \), where \( C \) is the constant of integration. This simplifies to \( \frac{5}{3} x^{3} + C \). Integrating is like having a treasure hunt where you're looking for the hidden values of \( C \). Integrals often represent the area under curves, and here, you're finding a new polynomial whose slope at any point reflects the original function, making it invaluable in calculus!
