QuestionDecember 14, 2025

4) Evaluate the indefinite intergal int(x^-(1)/(2)) d x

4) Evaluate the indefinite intergal int(x^-(1)/(2)) d x
4) Evaluate the indefinite intergal int(x^-(1)/(2)) d x

Solution
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Answer

2\sqrt{x} + C Explanation 1. Apply the power rule for integration Use **\int x^n \, dx = \frac{x^{n+1}}{n+1} + C**, valid for n \neq -1. 2. Substitute n = -\frac{1}{2} \int x^{-\frac{1}{2}} dx = \frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} + C = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C 3. Simplify \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}} = 2\sqrt{x}

Explanation

1. Apply the power rule for integration <br /> Use **$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$**, valid for $n \neq -1$.<br /><br />2. Substitute $n = -\frac{1}{2}$ <br /> $\int x^{-\frac{1}{2}} dx = \frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1} + C = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C$<br /><br />3. Simplify <br /> $\frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}} = 2\sqrt{x}$
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