Calculus Examples

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

Step 1

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $x + 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Substitute the lower limit in for $x$ in $u = x + 1$ .

$u_{lower} = 0 + 1$

Step 1.3

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = x + 1$ .

$u_{upper} = 1 + 1$

Step 1.5

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.

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

Expand $(u - 1) u^{- \frac{1}{2}}$ .

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Subtract $1$ from $2$ .

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

Step 4

Rewrite $- 1 u^{- \frac{1}{2}}$ as $- u^{- \frac{1}{2}}$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 8

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

$\frac{2}{3} u^{\frac{3}{2}}]_{1}^{2} - (2 u^{\frac{1}{2}}]_{1}^{2})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $2$ and at $1$ .

$(\frac{2}{3} \cdot 2^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}} - (2 u^{\frac{1}{2}}]_{1}^{2})$

Step 9.2

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

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

Step 9.3

Simplify.

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Step 9.3.1

Combine $\frac{2}{3}$ and $2^{\frac{3}{2}}$ .

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

Step 9.3.2

Multiply $2$ by $2^{\frac{3}{2}}$ by adding the exponents.

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Step 9.3.2.1

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

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Step 9.3.2.1.1

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

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

Step 9.3.2.1.2

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

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

Step 9.3.2.2

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

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

Step 9.3.2.3

Combine the numerators over the common denominator.

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

Step 9.3.2.4

Add $2$ and $3$ .

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}} - (2 \cdot 2^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.3

One to any power is one.

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} \cdot 1 - (2 \cdot 2^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.4

Multiply $- 1$ by $1$ .

$\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3} - (2 \cdot 2^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.5

Combine the numerators over the common denominator.

$\frac{2^{\frac{5}{2}} - 2}{3} - (2 \cdot 2^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.6

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

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Step 9.3.6.1

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

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Step 9.3.6.1.1

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

$\frac{2^{\frac{5}{2}} - 2}{3} - (2^{1} \cdot 2^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.6.1.2

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

$\frac{2^{\frac{5}{2}} - 2}{3} - (2^{1 + \frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.6.2

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

$\frac{2^{\frac{5}{2}} - 2}{3} - (2^{\frac{2}{2} + \frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.6.3

Combine the numerators over the common denominator.

$\frac{2^{\frac{5}{2}} - 2}{3} - (2^{\frac{2 + 1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.6.4

Add $2$ and $1$ .

$\frac{2^{\frac{5}{2}} - 2}{3} - (2^{\frac{3}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 9.3.7

One to any power is one.

$\frac{2^{\frac{5}{2}} - 2}{3} - (2^{\frac{3}{2}} - 2 \cdot 1)$

Step 9.3.8

Multiply $- 2$ by $1$ .

$\frac{2^{\frac{5}{2}} - 2}{3} - (2^{\frac{3}{2}} - 2)$

Step 9.3.9

To write $- (2^{\frac{3}{2}} - 2)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2^{\frac{5}{2}} - 2}{3} - (2^{\frac{3}{2}} - 2) \cdot \frac{3}{3}$

Step 9.3.10

Combine $- (2^{\frac{3}{2}} - 2)$ and $\frac{3}{3}$ .

$\frac{2^{\frac{5}{2}} - 2}{3} + \frac{- (2^{\frac{3}{2}} - 2) \cdot 3}{3}$

Step 9.3.11

Combine the numerators over the common denominator.

$\frac{2^{\frac{5}{2}} - 2 - (2^{\frac{3}{2}} - 2) \cdot 3}{3}$

Step 9.3.12

Multiply $3$ by $- 1$ .

$\frac{2^{\frac{5}{2}} - 2 - 3 (2^{\frac{3}{2}} - 2)}{3}$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{2^{\frac{5}{2}} - 2 - 3 \cdot 2^{\frac{3}{2}} - 3 \cdot - 2}{3}$

Step 10.2

Multiply $- 3$ by $- 2$ .

$\frac{2^{\frac{5}{2}} - 2 - 3 \cdot 2^{\frac{3}{2}} + 6}{3}$

Step 10.3

Add $- 2$ and $6$ .

$\frac{2^{\frac{5}{2}} + 4 - 3 \cdot 2^{\frac{3}{2}}}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{\frac{5}{2}} + 4 - 3 \cdot 2^{\frac{3}{2}}}{3}$

Decimal Form:

$0.39052429 \dots$

Step 12