Calculus Examples

$\int_{\sqrt{x}}^{\frac{π}{6}} θ \tan (θ) d θ$

Step 1

Swap the bounds of integration.

$\frac{d}{d x} [- \int_{\frac{π}{6}}^{\sqrt{x}} θ \tan (θ) d θ]$

Step 2

Take the derivative of $- \int_{\frac{π}{6}}^{\sqrt{x}} θ \tan (θ) d θ$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

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

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (- (\sqrt{x} \tan (\sqrt{x})))$

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

Combine $\frac{1}{2 x^{\frac{1}{2}}}$ and $\sqrt{x}$ .

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

Step 11

Combine $\tan (\sqrt{x})$ and $\frac{\sqrt{x}}{2 x^{\frac{1}{2}}}$ .

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