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tweety1234
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\int_0^1 \sqrt{x} - x^3 dx
How do I evaluate this?
what does \sqrt{x} -x^3 = ?
How do I evaluate this?
what does \sqrt{x} -x^3 = ?
Evaluating Definite Integrals: \int_0^1 \sqrt{x} - x^3 dx Explained
- Thread starter tweety1234
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SUMMARY
The integral \(\int_0^1 \sqrt{x} - x^3 \, dx\) can be evaluated using the power rule for integration. The expression simplifies to \(\int_0^1 x^{1/2} \, dx - \int_0^1 x^3 \, dx\). Applying the power rule, the integral of \(x^{1/2}\) results in \(\frac{2}{3} x^{3/2}\) evaluated from 0 to 1, yielding \(\frac{2}{3}\). The integral of \(x^3\) results in \(\frac{1}{4} x^4\) evaluated from 0 to 1, yielding \(\frac{1}{4}\). Thus, the final result is \(\frac{2}{3} - \frac{1}{4} = \frac{5}{12}\).
PREREQUISITES- Understanding of definite integrals
- Familiarity with the power rule of integration
- Basic knowledge of algebraic manipulation
- Ability to evaluate limits of integration
- Study the power rule for integration in detail
- Learn about techniques for evaluating definite integrals
- Explore applications of definite integrals in real-world scenarios
- Review common algebraic manipulations used in calculus
Students of calculus, mathematics educators, and anyone looking to strengthen their understanding of integral evaluation techniques.