Calculus Examples

$\int \frac{1}{\sqrt{x} \sqrt{1 - x}} d x$

Step 1

Combine using the product rule for radicals.

$\frac{d}{d x} [\int \frac{1}{\sqrt{x (1 - x)}} d x]$

Step 2

Apply basic rules of exponents.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x (1 - x)}$ as ${(x (1 - x))}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\int \frac{1}{{(x (1 - x))}^{\frac{1}{2}}} d x]$

Step 2.2

$\int \frac{1}{{(x (1 - x))}^{\frac{1}{2}}} d x$ is an antiderivative of $\frac{1}{{(x (1 - x))}^{\frac{1}{2}}}$ , so by definition $\frac{d}{d x} [\int \frac{1}{{(x (1 - x))}^{\frac{1}{2}}} d x]$ is $\frac{1}{{(x (1 - x))}^{\frac{1}{2}}}$ .

$\frac{1}{{(x (1 - x))}^{\frac{1}{2}}}$

Step 2.3

Apply the product rule to $x (1 - x)$ .

$\frac{1}{x^{\frac{1}{2}} {(1 - x)}^{\frac{1}{2}}}$