Calculus Examples

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

Step 1

Swap the bounds of integration.

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

Step 2

Take the derivative of $- \int_{2 π}^{\sqrt[3]{x}} \cos (t^{3}) d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [\sqrt[3]{x}] (- \cos ({\sqrt[3]{x}}^{3}))$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} - 1} (- \cos ({\sqrt[3]{x}}^{3}))$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (- \cos ({\sqrt[3]{x}}^{3}))$

Step 5

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (- \cos ({\sqrt[3]{x}}^{3}))$

Step 6

Combine the numerators over the common denominator.

$\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}} (- \cos ({\sqrt[3]{x}}^{3}))$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} x^{\frac{1 - 3}{3}} (- \cos ({\sqrt[3]{x}}^{3}))$

Step 7.2

Subtract $3$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

Simplify.

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Step 9.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}} (- \cos ({\sqrt[3]{x}}^{3}))$

Step 9.2

Multiply $\frac{1}{3}$ by $\frac{1}{x^{\frac{2}{3}}}$ .

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

Step 10

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\frac{1}{3 x^{\frac{2}{3}}} (- \cos ({(x^{\frac{1}{3}})}^{3}))$

Step 10.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{3 x^{\frac{2}{3}}} (- \cos (x^{\frac{1}{3} \cdot 3}))$

Step 10.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{1}{3 x^{\frac{2}{3}}} (- \cos (x^{\frac{3}{3}}))$

Step 10.4

Cancel the common factor of $3$ .

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Step 10.4.1

Cancel the common factor.

$\frac{1}{3 x^{\frac{2}{3}}} (- \cos (x^{\frac{3}{3}}))$

Step 10.4.2

Rewrite the expression.

$\frac{1}{3 x^{\frac{2}{3}}} (- \cos (x^{1}))$

Step 10.5

Simplify.

$\frac{1}{3 x^{\frac{2}{3}}} (- \cos (x))$

Step 11

Combine $\frac{1}{3 x^{\frac{2}{3}}}$ and $\cos (x)$ .

$- \frac{\cos (x)}{3 x^{\frac{2}{3}}}$