Calculus Examples

$\int_{0}^{2} \frac{x}{\sqrt{1 + 2 x^{2}}} d x$

Step 1

Let $u = 1 + 2 x^{2}$ . Then $d u = 4 x d x$ , so $\frac{1}{4} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 + 2 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 + 2 x^{2}$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 + 2 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [2 x^{2}]$ .

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

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [2 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x^{2}]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{2}$ with respect to $x$ is $2 \frac{d}{d x} [x^{2}]$ .

$0 + 2 \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

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

$0 + 2 (2 x)$

Step 1.1.3.3

Multiply $2$ by $2$ .

$0 + 4 x$

Step 1.1.4

Add $0$ and $4 x$ .

$4 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 + 2 x^{2}$ .

$u_{lower} = 1 + 2 \cdot 0^{2}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 1 + 2 \cdot 0$

Step 1.3.1.2

Multiply $2$ by $0$ .

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 + 2 x^{2}$ .

$u_{upper} = 1 + 2 \cdot 2^{2}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 1.5.1.1.1

Multiply $2$ by $2^{2}$ .

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Step 1.5.1.1.1.1

Raise $2$ to the power of $1$ .

$u_{upper} = 1 + 2^{1} \cdot 2^{2}$

Step 1.5.1.1.1.2

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

$u_{upper} = 1 + 2^{1 + 2}$

Step 1.5.1.1.2

Add $1$ and $2$ .

$u_{upper} = 1 + 2^{3}$

Step 1.5.1.2

Raise $2$ to the power of $3$ .

$u_{upper} = 1 + 8$

Step 1.5.2

Add $1$ and $8$ .

$u_{upper} = 9$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 9$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{9} \frac{1}{\sqrt{u}} \cdot \frac{1}{4} d u$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{4}$ .

$\int_{1}^{9} \frac{1}{\sqrt{u} \cdot 4} d u$

Step 2.2

Move $4$ to the left of $\sqrt{u}$ .

$\int_{1}^{9} \frac{1}{4 \sqrt{u}} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{1}^{9} \frac{1}{\sqrt{u}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$\frac{1}{4} \int_{1}^{9} \frac{1}{u^{\frac{1}{2}}} d u$

Step 4.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{4} \int_{1}^{9} {(u^{\frac{1}{2}})}^{- 1} d u$

Step 4.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 4.3.1

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

$\frac{1}{4} \int_{1}^{9} u^{\frac{1}{2} \cdot - 1} d u$

Step 4.3.2

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

$\frac{1}{4} \int_{1}^{9} u^{\frac{- 1}{2}} d u$

Step 4.3.3

Move the negative in front of the fraction.

$\frac{1}{4} \int_{1}^{9} u^{- \frac{1}{2}} d u$

Step 5

By the Power Rule, the integral of $u^{- \frac{1}{2}}$ with respect to $u$ is $2 u^{\frac{1}{2}}$ .

$\frac{1}{4} 2 u^{\frac{1}{2}}]_{1}^{9}$

Step 6

Simplify the expression.

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Step 6.1

Evaluate $2 u^{\frac{1}{2}}$ at $9$ and at $1$ .

$\frac{1}{4} ((2 \cdot 9^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{1}{4} (2 \cdot {(3^{2})}^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.2.2

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

$\frac{1}{4} (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Cancel the common factor.

$\frac{1}{4} (2 \cdot 3^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.2.3.2

Rewrite the expression.

$\frac{1}{4} (2 \cdot 3^{1} - 2 \cdot 1^{\frac{1}{2}})$

Step 6.2.4

Evaluate the exponent.

$\frac{1}{4} (2 \cdot 3 - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3

Simplify the expression.

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Step 6.3.1

Multiply $2$ by $3$ .

$\frac{1}{4} (6 - 2 \cdot 1^{\frac{1}{2}})$

Step 6.3.2

One to any power is one.

$\frac{1}{4} (6 - 2 \cdot 1)$

Step 6.3.3

Multiply $- 2$ by $1$ .

$\frac{1}{4} (6 - 2)$

Step 6.3.4

Subtract $2$ from $6$ .

$\frac{1}{4} \cdot 4$

Step 6.4

Simplify.

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Step 6.4.1

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

$\frac{4}{4}$

Step 6.4.2

Cancel the common factor of $4$ .

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Step 6.4.2.1

Cancel the common factor.

$\frac{4}{4}$

Step 6.4.2.2

Rewrite the expression.

$1$