Calculus Examples

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

Step 1

Let $u = x^{2} - 3$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{2} - 3$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} - 3$ .

$\frac{d}{d x} [x^{2} - 3]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]$

Step 1.1.3

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

$2 x + \frac{d}{d x} [- 3]$

Step 1.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{2} - 3$ .

$u_{lower} = 2^{2} - 3$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 4 - 3$

Step 1.3.2

Subtract $3$ from $4$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{2} - 3$ .

$u_{upper} = 3^{2} - 3$

Step 1.5

Simplify.

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Step 1.5.1

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

$u_{upper} = 9 - 3$

Step 1.5.2

Subtract $3$ from $9$ .

$u_{upper} = 6$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 6$

Step 1.7

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

$\int_{1}^{6} \frac{1}{u^{2}} \cdot \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

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

$\int_{1}^{6} \frac{1}{u^{2} \cdot 2} d u$

Step 2.2

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

$\int_{1}^{6} \frac{1}{2 u^{2}} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{1}^{6} \frac{1}{u^{2}} d u$

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.

$\frac{1}{2} \int_{1}^{6} {(u^{2})}^{- 1} d u$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} \int_{1}^{6} u^{2 \cdot - 1} d u$

Step 4.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{2} \int_{1}^{6} u^{- 2} d u$

Step 5

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

$\frac{1}{2} - u^{- 1}]_{1}^{6}$

Step 6

Evaluate $- u^{- 1}$ at $6$ and at $1$ .

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

Step 7

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

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

Step 8

One to any power is one.

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{- 1 + 6}{6}$

Step 9.3

Add $- 1$ and $6$ .

$\frac{1}{2} \cdot \frac{5}{6}$

Step 10

Simplify.

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Step 10.1

Multiply $\frac{1}{2}$ by $\frac{5}{6}$ .

$\frac{5}{2 \cdot 6}$

Step 10.2

Multiply $2$ by $6$ .

$\frac{5}{12}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{12}$

Decimal Form:

$0.41\overline{6}$