Gough Estrada
03/27/2023 · Middle School
Obtener la siguiente integral definida: \( \int_{0}^{1}\left(\frac{x^{8}+x^{4}+x^{\frac{13}{3}}}{x^{4}}\right) d x \)
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Step-by-step Solution
Calculate the integral \( \int_{0}^{1}\left(\frac{x^{8}+x^{4}+x^{\frac{13}{3}}}{x^{4}}\right) dx \).
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
\(\int_{0}^{1} \left(\frac{x^{8}+x^{4}+x^{\frac{13}{3}}}{x^{4}}\right) dx\)
- step1: Evaluate:
\(\int_{0}^{1} \frac{x^{8}+x^{4}+x^{\frac{13}{3}}}{x^{4}} dx\)
- step2: Simplify the expression:
\(\int_{0}^{1} x^{4}+1+x^{\frac{1}{3}} dx\)
- step3: Rewrite the improper integral:
\(\lim _{a\rightarrow 0^{+}}\left(\int_{a}^{1} x^{4}+1+x^{\frac{1}{3}} dx\right)\)
- step4: Evaluate the integral:
\(\lim _{a\rightarrow 0^{+}}\left(\frac{39}{20}-\frac{1}{5}a^{5}-a-\frac{3}{4}a^{\frac{4}{3}}\right)\)
- step5: Rewrite the expression:
\(\lim _{a\rightarrow 0^{+}}\left(\frac{39}{20}\right)-\lim _{a\rightarrow 0^{+}}\left(\frac{1}{5}a^{5}\right)-\lim _{a\rightarrow 0^{+}}\left(a\right)-\lim _{a\rightarrow 0^{+}}\left(\frac{3}{4}a^{\frac{4}{3}}\right)\)
- step6: Calculate:
\(\frac{39}{20}-0-0-0\)
- step7: Calculate:
\(\frac{39}{20}-0-0\)
- step8: Calculate:
\(\frac{39}{20}-0\)
- step9: Calculate:
\(\frac{39}{20}\)
La integral definida \( \int_{0}^{1}\left(\frac{x^{8}+x^{4}+x^{\frac{13}{3}}}{x^{4}}\right) dx \) es igual a \( \frac{39}{20} \) o \( 1\frac{19}{20} \) o \( 1.95 \).
Quick Answer
La integral es \( \frac{39}{20} \) o \( 1\frac{19}{20} \) o \( 1.95 \).
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