Calculus Examples

\int_{- 2}^{2} (x^{3} cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - x^{2} d x} d x

$\int_{- 2}^{2} (x^{3} \cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - x^{2} d x} d x$

Step 1

Raise

x

$x$ to the power of

1

$1$ .

\frac{d}{d x} [\int_{- 2}^{2} (x^{3} cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - 1 d (x^{2} x^{1})} d x]

$\frac{d}{d x} [\int_{- 2}^{2} (x^{3} \cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - 1 d (x^{2} x^{1})} d x]$

Step 2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{d}{d x} [\int_{- 2}^{2} (x^{3} cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - 1 d x^{2 + 1}} d x]

$\frac{d}{d x} [\int_{- 2}^{2} (x^{3} \cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - 1 d x^{2 + 1}} d x]$

Step 3

Simplify the expression.

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Step 3.1

Add

2

$2$ and

1

$1$ .

\frac{d}{d x} [\int_{- 2}^{2} (x^{3} cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - 1 d x^{3}} d x]

$\frac{d}{d x} [\int_{- 2}^{2} (x^{3} \cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - 1 d x^{3}} d x]$

Step 3.2

Rewrite

- 1 d

$- 1 d$ as

- d

$- d$ .

\frac{d}{d x} [\int_{- 2}^{2} (x^{3} cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - d x^{3}} d x]

$\frac{d}{d x} [\int_{- 2}^{2} (x^{3} \cos (\frac{x}{2}) + \frac{1}{2}) \sqrt{4 - d x^{3}} d x]$

Step 3.3

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{4 - d x^{3}}

$\sqrt{4 - d x^{3}}$ as

{(4 - d x^{3})}^{\frac{1}{2}}

${(4 - d x^{3})}^{\frac{1}{2}}$ .

\frac{d}{d x} [\int_{- 2}^{2} (x^{3} cos (\frac{x}{2}) + \frac{1}{2}) {(4 - d x^{3})}^{\frac{1}{2}} d x]

$\frac{d}{d x} [\int_{- 2}^{2} (x^{3} \cos (\frac{x}{2}) + \frac{1}{2}) {(4 - d x^{3})}^{\frac{1}{2}} d x]$

Step 4

Once

$\int_{- 2}^{2} (x^{3} \cos (\frac{x}{2}) + \frac{1}{2}) {(4 - d x^{3})}^{\frac{1}{2}} d x$ has been evaluated, it will be constant with respect to

$x$ , so the derivative of

$\int_{- 2}^{2} (x^{3} \cos (\frac{x}{2}) + \frac{1}{2}) {(4 - d x^{3})}^{\frac{1}{2}} d x$ with respect to

$x$ is

$0$ .

$0$