Calculus Examples

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

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.

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

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{u^{2}}$ by $\frac{1}{4}$ .

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

Step 2.2

Move $4$ to the left of $u^{2}$ .

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $- 1$ .

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

Step 5

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

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

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $- u^{- 1}$ at $9$ and at $1$ .

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

One to any power is one.

$\frac{1}{4} (- \frac{1}{9} + 1)$

Step 6.2.3

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

$\frac{1}{4} (- \frac{1}{9} + \frac{9}{9})$

Step 6.2.4

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{- 1 + 9}{9}$

Step 6.2.5

Add $- 1$ and $9$ .

$\frac{1}{4} \cdot \frac{8}{9}$

Step 6.2.6

Multiply $\frac{1}{4}$ by $\frac{8}{9}$ .

$\frac{8}{4 \cdot 9}$

Step 6.2.7

Multiply $4$ by $9$ .

$\frac{8}{36}$

Step 6.2.8

Cancel the common factor of $8$ and $36$ .

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Step 6.2.8.1

Factor $4$ out of $8$ .

$\frac{4 (2)}{36}$

Step 6.2.8.2

Cancel the common factors.

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Step 6.2.8.2.1

Factor $4$ out of $36$ .

$\frac{4 \cdot 2}{4 \cdot 9}$

Step 6.2.8.2.2

Cancel the common factor.

$\frac{4 \cdot 2}{4 \cdot 9}$

Step 6.2.8.2.3

Rewrite the expression.

$\frac{2}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{9}$

Decimal Form:

$0.\overline{2}$

Step 8