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La integral de \( \int\left(x^{2}+3\right)^{2} dx \) es \( \frac{x^{5}}{5}+2x^{3}+9x + C \).
Solution
Calculate the integral \( \int\left(x^{2}+3\right)^{2} dx \).
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
\(\int \left(x^{2}+3\right)^{2} dx\)
- step1: Expand the expression:
\(\int x^{4}+6x^{2}+9 dx\)
- step2: Use properties of integrals:
\(\int x^{4} dx+\int 6x^{2} dx+\int 9 dx\)
- step3: Evaluate the integral:
\(\frac{x^{5}}{5}+\int 6x^{2} dx+\int 9 dx\)
- step4: Evaluate the integral:
\(\frac{x^{5}}{5}+2x^{3}+\int 9 dx\)
- step5: Evaluate the integral:
\(\frac{x^{5}}{5}+2x^{3}+9x\)
- step6: Add the constant of integral C:
\(\frac{x^{5}}{5}+2x^{3}+9x + C, C \in \mathbb{R}\)
La integral de \( \int\left(x^{2}+3\right)^{2} dx \) es igual a \( \frac{x^{5}}{5}+2x^{3}+9x + C \), donde \( C \) es una constante real.
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Simplify this solution The Deep Dive
To solve the integral \( \int\left(x^{2}+3\right)^{2} d x \), we can start by expanding the integrand. 1. Expanding the expression: \[ (x^{2} + 3)^{2} = x^{4} + 6x^{2} + 9. \] 2. Now, integrate term by term: \[ \int (x^{4} + 6x^{2} + 9) dx = \int x^{4} dx + \int 6x^{2} dx + \int 9 dx. \] 3. Carrying out the integrations: \[ \frac{x^{5}}{5} + 6 \cdot \frac{x^{3}}{3} + 9x + C = \frac{x^{5}}{5} + 2x^{3} + 9x + C, \] where \( C \) is the constant of integration. Therefore, the final result is: \[ \int\left(x^{2}+3\right)^{2} d x = \frac{x^{5}}{5} + 2x^{3} + 9x + C. \]
