Pregunta
\( \int _{}^{}15(x+2)^{4} d \)
Ask by Vaughan Macdonald. in Mexico
Feb 27,2025
Solución de inteligencia artificial de Upstudy
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The integral of 15(x + 2)⁴ with respect to x is 3(x + 2)⁵ plus a constant.
Solución
To solve the integral ∫15(x + 2)⁴ dx, we can use a substitution:
1. Let u = x + 2, then du = dx.
2. Substitute into the integral: ∫15u⁴ du.
3. Integrate: 15 ∫u⁴ du = 15 · (u⁵/5) = 3u⁵.
4. Substitute back the original variable: 3(x + 2)⁵.
Thus, the antiderivative is 3(x + 2)⁵ + C, where C is the constant of integration.
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Simplifique esta solución Mind Expander
To solve the integral \( \int 15(x+2)^{4} dx \), you can use the power rule for integrals. The integral of \( (x+a)^{n} \) is given by \( \frac{(x+a)^{n+1}}{n+1} + C \). Here, \( a = 2 \) and \( n = 4 \). So, you will get: \[ \int 15(x+2)^{4} dx = 15 \cdot \frac{(x+2)^{4+1}}{4+1} + C = 15 \cdot \frac{(x+2)^{5}}{5} + C = 3(x+2)^{5} + C. \] Now, don't forget to add the constant \( C \) to represent the family of antiderivatives! Furthermore, it’s always good to remember to double-check your work by differentiating your result to ensure you arrive back at the original integrand.
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