Calculus Examples

$\int_{1}^{4} \sqrt{x} \ln (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = \sqrt{x}$ .

$\ln (x) (\frac{2}{3} x^{\frac{3}{2}})]_{1}^{4} - \int_{1}^{4} \frac{2}{3} x^{\frac{3}{2}} \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

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

$\ln (x) \frac{2 x^{\frac{3}{2}}}{3}]_{1}^{4} - \int_{1}^{4} \frac{2}{3} x^{\frac{3}{2}} \frac{1}{x} d x$

Step 2.2

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

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - \int_{1}^{4} \frac{2}{3} x^{\frac{3}{2}} \frac{1}{x} d x$

Step 3

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - (\frac{2}{3} \int_{1}^{4} x^{\frac{3}{2}} \frac{1}{x} d x)$

Step 4

Simplify.

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Step 4.1

Combine $x^{\frac{3}{2}}$ and $\frac{1}{x}$ .

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - (\frac{2}{3} \int_{1}^{4} \frac{x^{\frac{3}{2}}}{x} d x)$

Step 4.2

Move $x^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - (\frac{2}{3} \int_{1}^{4} x^{\frac{3}{2}} x^{- 1} d x)$

Step 4.3

Multiply $x^{\frac{3}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 4.3.1

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

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - (\frac{2}{3} \int_{1}^{4} x^{\frac{3}{2} - 1} d x)$

Step 4.3.2

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

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - (\frac{2}{3} \int_{1}^{4} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}} d x)$

Step 4.3.3

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

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - (\frac{2}{3} \int_{1}^{4} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} d x)$

Step 4.3.4

Combine the numerators over the common denominator.

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - (\frac{2}{3} \int_{1}^{4} x^{\frac{3 - 1 \cdot 2}{2}} d x)$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $- 1$ by $2$ .

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - (\frac{2}{3} \int_{1}^{4} x^{\frac{3 - 2}{2}} d x)$

Step 4.3.5.2

Subtract $2$ from $3$ .

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - (\frac{2}{3} \int_{1}^{4} x^{\frac{1}{2}} d x)$

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - \frac{2}{3} \int_{1}^{4} x^{\frac{1}{2}} d x$

Step 5

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

$\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}]_{1}^{4} - \frac{2}{3} \frac{2}{3} x^{\frac{3}{2}}]_{1}^{4}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{\ln (x) (2 x^{\frac{3}{2}})}{3}$ at $4$ and at $1$ .

$(\frac{\ln (4) (2 \cdot 4^{\frac{3}{2}})}{3}) - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} \frac{2}{3} x^{\frac{3}{2}}]_{1}^{4}$

Step 6.2

Evaluate $\frac{2}{3} x^{\frac{3}{2}}$ at $4$ and at $1$ .

$(\frac{\ln (4) (2 \cdot 4^{\frac{3}{2}})}{3}) - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3

Simplify.

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Step 6.3.1

Rewrite $4$ as $2^{2}$ .

$\frac{\ln (4) (2 \cdot {(2^{2})}^{\frac{3}{2}})}{3} - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.2

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

$\frac{\ln (4) (2 \cdot 2^{2 (\frac{3}{2})})}{3} - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.3

Cancel the common factor of $2$ .

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Step 6.3.3.1

Cancel the common factor.

$\frac{\ln (4) (2 \cdot 2^{2 (\frac{3}{2})})}{3} - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.3.2

Rewrite the expression.

$\frac{\ln (4) (2 \cdot 2^{3})}{3} - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.4

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

$\frac{\ln (4) (2 \cdot 8)}{3} - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.5

Multiply $2$ by $8$ .

$\frac{\ln (4) \cdot 16}{3} - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.6

Move $16$ to the left of $\ln (4)$ .

$\frac{16 \cdot \ln (4)}{3} - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.7

One to any power is one.

$\frac{16 \ln (4)}{3} - \frac{\ln (1) (2 \cdot 1)}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.8

Multiply $2$ by $1$ .

$\frac{16 \ln (4)}{3} - \frac{\ln (1) \cdot 2}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.9

Move $2$ to the left of $\ln (1)$ .

$\frac{16 \ln (4)}{3} - \frac{2 \cdot \ln (1)}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 4^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.10

Rewrite $4$ as $2^{2}$ .

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{2}{3} \cdot {(2^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.11

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

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{2}{3} \cdot 2^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.12

Cancel the common factor of $2$ .

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Step 6.3.12.1

Cancel the common factor.

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{2}{3} \cdot 2^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.12.2

Rewrite the expression.

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{2}{3} \cdot 2^{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.13

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

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{2}{3} \cdot 8 - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.14

Combine $\frac{2}{3}$ and $8$ .

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{2 \cdot 8}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.15

Multiply $2$ by $8$ .

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{16}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.3.16

One to any power is one.

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{16}{3} - \frac{2}{3} \cdot 1)$

Step 6.3.17

Multiply $- 1$ by $1$ .

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{16}{3} - \frac{2}{3})$

Step 6.3.18

Combine the numerators over the common denominator.

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} \cdot \frac{16 - 2}{3}$

Step 6.3.19

Subtract $2$ from $16$ .

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} \cdot \frac{14}{3}$

Step 6.3.20

Multiply $\frac{14}{3}$ by $\frac{2}{3}$ .

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{14 \cdot 2}{3 \cdot 3}$

Step 6.3.21

Multiply $14$ by $2$ .

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{28}{3 \cdot 3}$

Step 6.3.22

Multiply $3$ by $3$ .

$\frac{16 \ln (4)}{3} - \frac{2 \ln (1)}{3} - \frac{28}{9}$

Step 7

Simplify.

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Step 7.1

Combine the numerators over the common denominator.

$\frac{16 \ln (4) - 2 \ln (1)}{3} + \frac{- 28}{9}$

Step 7.2

Simplify each term.

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Step 7.2.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$\frac{16 \ln (2^{2}) - 2 \ln (1)}{3} + \frac{- 28}{9}$

Step 7.2.2

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

$\frac{16 (2 \ln (2)) - 2 \ln (1)}{3} + \frac{- 28}{9}$

Step 7.2.3

Multiply $2$ by $16$ .

$\frac{32 \ln (2) - 2 \ln (1)}{3} + \frac{- 28}{9}$

Step 7.2.4

The natural logarithm of $1$ is $0$ .

$\frac{32 \ln (2) - 2 \cdot 0}{3} + \frac{- 28}{9}$

Step 7.2.5

Multiply $- 2$ by $0$ .

$\frac{32 \ln (2) + 0}{3} + \frac{- 28}{9}$

Step 7.3

Add $32 \ln (2)$ and $0$ .

$\frac{32 \ln (2)}{3} + \frac{- 28}{9}$

Step 7.4

Move the negative in front of the fraction.

$\frac{32 \ln (2)}{3} - \frac{28}{9}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{32 \ln (2)}{3} - \frac{28}{9}$

Decimal Form:

$4.28245881 \dots$