Calculus Examples

$\int_{0}^{1} \frac{x}{{(x^{2} + 1)}^{3}} d x$

Step 1

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

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Step 1.1.1

Differentiate $x^{2} + 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ 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 $u = x^{2} + 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 0 + 1$

Step 1.3.2

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 1.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 1.7

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

$\int_{1}^{2} \frac{1}{u^{3}} \cdot \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{u^{3}}$ by $\frac{1}{2}$ .

$\int_{1}^{2} \frac{1}{u^{3} \cdot 2} d u$

Step 2.2

Move $2$ to the left of $u^{3}$ .

$\int_{1}^{2} \frac{1}{2 u^{3}} d u$

Step 3

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

$\frac{1}{2} \int_{1}^{2} \frac{1}{u^{3}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $u^{3}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{2} \int_{1}^{2} {(u^{3})}^{- 1} d u$

Step 4.2

Multiply the exponents in ${(u^{3})}^{- 1}$ .

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Step 4.2.1

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

$\frac{1}{2} \int_{1}^{2} u^{3 \cdot - 1} d u$

Step 4.2.2

Multiply $3$ by $- 1$ .

$\frac{1}{2} \int_{1}^{2} u^{- 3} d u$

Step 5

By the Power Rule, the integral of $u^{- 3}$ with respect to $u$ is $- \frac{1}{2} u^{- 2}$ .

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

Multiply $4$ by $2$ .

$\frac{1}{2} (- \frac{1}{8} + \frac{1}{2} \cdot 1^{- 2})$

Step 6.2.5

One to any power is one.

$\frac{1}{2} (- \frac{1}{8} + \frac{1}{2} \cdot 1)$

Step 6.2.6

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

$\frac{1}{2} (- \frac{1}{8} + \frac{1}{2})$

Step 6.2.7

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{2} (- \frac{1}{8} + \frac{1}{2} \cdot \frac{4}{4})$

Step 6.2.8

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.8.1

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

$\frac{1}{2} (- \frac{1}{8} + \frac{4}{2 \cdot 4})$

Step 6.2.8.2

Multiply $2$ by $4$ .

$\frac{1}{2} (- \frac{1}{8} + \frac{4}{8})$

Step 6.2.9

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{- 1 + 4}{8}$

Step 6.2.10

Add $- 1$ and $4$ .

$\frac{1}{2} \cdot \frac{3}{8}$

Step 6.2.11

Multiply $\frac{1}{2}$ by $\frac{3}{8}$ .

$\frac{3}{2 \cdot 8}$

Step 6.2.12

Multiply $2$ by $8$ .

$\frac{3}{16}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{16}$

Decimal Form:

$0.1875$

Step 8