Calculus Examples

$\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4} + 1} d z$

Step 1

Take the derivative of $\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4} + 1} d z$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [\sqrt{x}] \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

$\frac{d}{d x} [x^{\frac{1}{2}}] \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 2.2

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

$\frac{1}{2} x^{\frac{1}{2} - 1} \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 3

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

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 4

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

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{\frac{1 - 2}{2}} \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 6.2

Subtract $2$ from $1$ .

$\frac{1}{2} x^{\frac{- 1}{2}} \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 7

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}} \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 8

Simplify.

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Step 8.1

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

$\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 8.2

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

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{{\sqrt{x}}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 9

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 9.1

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

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{{(x^{\frac{1}{2}})}^{2}}{{\sqrt{x}}^{4} + 1}$

Step 9.2

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

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x^{\frac{1}{2} \cdot 2}}{{\sqrt{x}}^{4} + 1}$

Step 9.3

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

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x^{\frac{2}{2}}}{{\sqrt{x}}^{4} + 1}$

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Cancel the common factor.

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x^{\frac{2}{2}}}{{\sqrt{x}}^{4} + 1}$

Step 9.4.2

Rewrite the expression.

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x^{1}}{{\sqrt{x}}^{4} + 1}$

Step 9.5

Simplify.

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x}{{\sqrt{x}}^{4} + 1}$

Step 10

Rewrite ${\sqrt{x}}^{4}$ as $x^{2}$ .

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Step 10.1

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

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x}{{(x^{\frac{1}{2}})}^{4} + 1}$

Step 10.2

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

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x}{x^{\frac{1}{2} \cdot 4} + 1}$

Step 10.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x}{x^{\frac{4}{2}} + 1}$

Step 10.4

Cancel the common factor of $4$ and $2$ .

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Step 10.4.1

Factor $2$ out of $4$ .

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x}{x^{\frac{2 \cdot 2}{2}} + 1}$

Step 10.4.2

Cancel the common factors.

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Step 10.4.2.1

Factor $2$ out of $2$ .

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

Step 10.4.2.2

Cancel the common factor.

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x}{x^{\frac{2 \cdot 2}{2 \cdot 1}} + 1}$

Step 10.4.2.3

Rewrite the expression.

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x}{x^{\frac{2}{1}} + 1}$

Step 10.4.2.4

Divide $2$ by $1$ .

$\frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{x}{x^{2} + 1}$

Step 11

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

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

Step 12

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 13

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 13.1

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

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Step 13.1.1

Raise $x$ to the power of $1$ .

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

Step 13.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 13.2

Write $1$ as a fraction with a common denominator.

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

Step 13.3

Combine the numerators over the common denominator.

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

Step 13.4

Subtract $1$ from $2$ .

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