Norris Zimmerman
07/24/2024 · Junior High School
4. \( \int \frac{1}{2}[10-(x / 2)]^{4} d x \)
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Tutor-Verified Answer
Step-by-step Solution
Calculate or simplify the expression \( \int \frac{1}{2}[10-(x / 2)]^{4} dx \).
Evaluate the integral by following steps:
- step0: Evaluate:
\(\int \frac{1}{2}\left(10-\frac{x}{2}\right)^{4} dx\)
- step1: Use properties of integrals:
\(\frac{1}{2}\times \int \left(10-\frac{x}{2}\right)^{4} dx\)
- step2: Evaluate the integral:
\(\frac{1}{2}\times \int 10000-2000x+150x^{2}-5x^{3}+\frac{x^{4}}{16} dx\)
- step3: Use properties of integrals:
\(\frac{1}{2}\left(\int 10000 dx-\int 2000x dx+\int 150x^{2} dx-\int 5x^{3} dx+\int \frac{x^{4}}{16} dx\right)\)
- step4: Evaluate the integral:
\(\frac{1}{2}\left(10000x-1000x^{2}+50x^{3}-\frac{5}{4}x^{4}+\frac{1}{80}x^{5}\right)\)
- step5: Evaluate the integral:
\(5000x-500x^{2}+25x^{3}-\frac{5}{8}x^{4}+\frac{1}{160}x^{5}\)
- step6: Add the constant of integral C:
\(5000x-500x^{2}+25x^{3}-\frac{5}{8}x^{4}+\frac{1}{160}x^{5} + C, C \in \mathbb{R}\)
La integral de la expresión dada es \(5000x-500x^{2}+25x^{3}-\frac{5}{8}x^{4}+\frac{1}{160}x^{5} + C\), donde \(C\) es una constante real.
Quick Answer
La integral es \(5000x-500x^{2}+25x^{3}-\frac{5}{8}x^{4}+\frac{1}{160}x^{5} + C\).
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