Stewart Schneider
09/10/2023 · Elementary School
Find the integral. \( \int \sqrt{\mathrm{x}^{3}} \mathrm{dx} \) \( \int \sqrt{\mathrm{x}^{3}} \mathrm{dx}=\square \)
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Step-by-step Solution
Calculate the integral \( \int \sqrt{x^3} dx \).
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
\(\int \sqrt{x^{3}} dx\)
- step1: Evaluate the power:
\(\int x^{\frac{3}{2}} dx\)
- step2: Evaluate the integral:
\(\frac{x^{\frac{5}{2}}}{\frac{5}{2}}\)
- step3: Simplify:
\(\frac{2}{5}x^{\frac{5}{2}}\)
- step4: Simplify:
\(\frac{2}{5}x^{2}\sqrt{x}\)
- step5: Add the constant of integral C:
\(\frac{2}{5}x^{2}\sqrt{x} + C, C \in \mathbb{R}\)
The integral \( \int \sqrt{x^3} dx \) is equal to \( \frac{2}{5}x^{2}\sqrt{x} + C \), where \( C \) is an arbitrary constant.
Quick Answer
\( \int \sqrt{x^3} dx = \frac{2}{5}x^{2}\sqrt{x} + C \)
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