Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $t + 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$u_{lower} = 1 + 1$

Step 1.3

Add $1$ and $1$ .

$u_{lower} = 2$

Step 1.4

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

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

Step 1.5

Add $1$ and $1$ .

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

Step 1.6

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

$u_{lower} = 2$

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

Step 1.7

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

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

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

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

Step 2.2

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

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

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

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

Step 2.3.2

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

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Subtract $1$ from $2$ .

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

Step 3.9

Multiply $2$ by $- 1$ .

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

Step 3.10

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

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

Step 3.11

Add $2 u^{\frac{1}{2}}$ and $0$ .

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

Step 4

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

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

Step 5

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

$2 (\frac{2}{3} u^{\frac{3}{2}}]_{2}^{x^{2} + 2})$

Step 6

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

$2 (\frac{2 u^{\frac{3}{2}}}{3}]_{2}^{x^{2} + 2})$

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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Step 7.2.1.1

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

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Step 7.2.1.1.1

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

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

Step 7.2.1.1.2

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

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

Step 7.2.1.2

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

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

Step 7.2.1.3

Combine the numerators over the common denominator.

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

Step 7.2.1.4

Add $2$ and $3$ .

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

Step 7.2.2

Combine the numerators over the common denominator.

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

Step 7.2.3

Combine $2$ and $\frac{2 {(x^{2} + 2)}^{\frac{3}{2}} - 2^{\frac{5}{2}}}{3}$ .

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

Step 8

Reorder terms.

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $\frac{2}{3} (2 {(x^{2} + 2)}^{\frac{3}{2}})$ .

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Step 9.2.1

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

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

Step 9.2.2

Multiply $2$ by $2$ .

$\frac{4}{3} {(x^{2} + 2)}^{\frac{3}{2}} + \frac{2}{3} (- 2^{\frac{5}{2}})$

Step 9.2.3

Combine $\frac{4}{3}$ and ${(x^{2} + 2)}^{\frac{3}{2}}$ .

$\frac{4 {(x^{2} + 2)}^{\frac{3}{2}}}{3} + \frac{2}{3} (- 2^{\frac{5}{2}})$

Step 9.3

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

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Step 9.3.1

Combine $\frac{2}{3}$ and $2^{\frac{5}{2}}$ .

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

Step 9.3.2

Multiply $2$ by $2^{\frac{5}{2}}$ by adding the exponents.

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Step 9.3.2.1

Multiply $2$ by $2^{\frac{5}{2}}$ .

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Step 9.3.2.1.1

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

$\frac{4 {(x^{2} + 2)}^{\frac{3}{2}}}{3} - \frac{2^{1} \cdot 2^{\frac{5}{2}}}{3}$

Step 9.3.2.1.2

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

$\frac{4 {(x^{2} + 2)}^{\frac{3}{2}}}{3} - \frac{2^{1 + \frac{5}{2}}}{3}$

Step 9.3.2.2

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

$\frac{4 {(x^{2} + 2)}^{\frac{3}{2}}}{3} - \frac{2^{\frac{2}{2} + \frac{5}{2}}}{3}$

Step 9.3.2.3

Combine the numerators over the common denominator.

$\frac{4 {(x^{2} + 2)}^{\frac{3}{2}}}{3} - \frac{2^{\frac{2 + 5}{2}}}{3}$

Step 9.3.2.4

Add $2$ and $5$ .

$\frac{4 {(x^{2} + 2)}^{\frac{3}{2}}}{3} - \frac{2^{\frac{7}{2}}}{3}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{4 {(x^{2} + 2)}^{\frac{3}{2}} - 2^{\frac{7}{2}}}{3}$