Calculus Examples

$\frac{d}{(d x) \int_{2 π}^{\sqrt[3]{x}} \cos (t) d t}$

Step 1

Cancel the common factor of $d$ .

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Step 1.1

Cancel the common factor.

$\frac{d}{d x \int_{2 π}^{\sqrt[3]{x}} \cos (t) d t}$

Step 1.2

Rewrite the expression.

$\frac{1}{x \int_{2 π}^{\sqrt[3]{x}} \cos (t) d t}$

Step 2

The integral of $\cos (t)$ with respect to $t$ is $\sin (t)$ .

$\frac{1}{x \sin (t)]_{2 π}^{\sqrt[3]{x}}}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\sin (t)$ at $\sqrt[3]{x}$ and at $2 π$ .

$\frac{1}{x (\sin (\sqrt[3]{x}) - \sin (2 π))}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the denominator.

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Step 3.2.1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{1}{x (\sin (\sqrt[3]{x}) - \sin (0))}$

Step 3.2.1.2

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{x (\sin (\sqrt[3]{x}) - 0)}$

Step 3.2.1.3

Multiply $- 1$ by $0$ .

$\frac{1}{x (\sin (\sqrt[3]{x}) + 0)}$

Step 3.2.1.4

Add $\sin (\sqrt[3]{x})$ and $0$ .

$\frac{1}{x \sin (\sqrt[3]{x})}$

Step 3.2.2

Separate fractions.

$\frac{1}{x} \cdot \frac{1}{\sin (\sqrt[3]{x})}$

Step 3.2.3

Convert from $\frac{1}{\sin (\sqrt[3]{x})}$ to $\csc (\sqrt[3]{x})$ .

$\frac{1}{x} \csc (\sqrt[3]{x})$

Step 3.2.4

Combine $\frac{1}{x}$ and $\csc (\sqrt[3]{x})$ .

$\frac{\csc (\sqrt[3]{x})}{x}$