Calculus Examples

$\int_{1}^{6} \sqrt{t} \ln (t) d t$

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

$\ln (t) (\frac{2}{3} t^{\frac{3}{2}})]_{1}^{6} - \int_{1}^{6} \frac{2}{3} t^{\frac{3}{2}} \frac{1}{t} d t$

Simplify.

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

$\ln (t) \frac{2 t^{\frac{3}{2}}}{3}]_{1}^{6} - \int_{1}^{6} \frac{2}{3} t^{\frac{3}{2}} \frac{1}{t} d t$

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

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2}{3} t^{\frac{3}{2}} \frac{1}{t} d t$

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

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 t^{\frac{3}{2}}}{3} \cdot \frac{1}{t} d t$

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

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 t^{\frac{3}{2}}}{3 t} d t$

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

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 t^{\frac{3}{2}} t^{- 1}}{3} d t$

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

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Move $t^{- 1}$ .

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 (t^{- 1} t^{\frac{3}{2}})}{3} d t$

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

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 t^{- 1 + \frac{3}{2}}}{3} d t$

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

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 t^{- 1 \cdot \frac{2}{2} + \frac{3}{2}}}{3} d t$

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

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 t^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}}}{3} d t$

Combine the numerators over the common denominator.

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 t^{\frac{- 1 \cdot 2 + 3}{2}}}{3} d t$

Simplify the numerator.

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Multiply $- 1$ by $2$ .

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 t^{\frac{- 2 + 3}{2}}}{3} d t$

Add $- 2$ and $3$ .

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \int_{1}^{6} \frac{2 t^{\frac{1}{2}}}{3} d t$

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

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - (\frac{2}{3} \int_{1}^{6} t^{\frac{1}{2}} d t)$

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

$\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}]_{1}^{6} - \frac{2}{3} \frac{2}{3} t^{\frac{3}{2}}]_{1}^{6}$

Substitute and simplify.

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Evaluate $\frac{\ln (t) (2 t^{\frac{3}{2}})}{3}$ at $6$ and at $1$ .

$(\frac{\ln (6) (2 \cdot 6^{\frac{3}{2}})}{3}) - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} \frac{2}{3} t^{\frac{3}{2}}]_{1}^{6}$

Evaluate $\frac{2}{3} t^{\frac{3}{2}}$ at $6$ and at $1$ .

$(\frac{\ln (6) (2 \cdot 6^{\frac{3}{2}})}{3}) - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 6^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Simplify.

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Move $2$ to the left of $\ln (6)$ .

$\frac{2 \cdot \ln (6) \cdot 6^{\frac{3}{2}}}{3} - \frac{\ln (1) (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 6^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

One to any power is one.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}}}{3} - \frac{\ln (1) (2 \cdot 1)}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 6^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Multiply $2$ by $1$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}}}{3} - \frac{\ln (1) \cdot 2}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 6^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

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

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}}}{3} - \frac{2 \cdot \ln (1)}{3} - \frac{2}{3} ((\frac{2}{3} \cdot 6^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Combine $\frac{2}{3}$ and $6^{\frac{3}{2}}$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}}}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

One to any power is one.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}}}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1)$

Multiply $- 1$ by $1$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}}}{3} - \frac{2 \ln (1)}{3} - \frac{2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})$

To write $- \frac{2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3}) \cdot \frac{3}{3} - \frac{2 \ln (1)}{3}$

Combine $- \frac{2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})$ and $\frac{3}{3}$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}}}{3} + \frac{- \frac{2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3}) \cdot 3}{3} - \frac{2 \ln (1)}{3}$

Combine the numerators over the common denominator.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} - \frac{2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3}) \cdot 3}{3} - \frac{2 \ln (1)}{3}$

Multiply $3$ by $- 1$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} - 3 (\frac{2}{3}) (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})}{3} - \frac{2 \ln (1)}{3}$

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

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{- 3 \cdot 2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})}{3} - \frac{2 \ln (1)}{3}$

Multiply $- 3$ by $2$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{- 6}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})}{3} - \frac{2 \ln (1)}{3}$

Cancel the common factor of $- 6$ and $3$ .

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Factor $3$ out of $- 6$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{3 \cdot - 2}{3} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})}{3} - \frac{2 \ln (1)}{3}$

Cancel the common factors.

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Factor $3$ out of $3$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{3 \cdot - 2}{3 (1)} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})}{3} - \frac{2 \ln (1)}{3}$

Cancel the common factor.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{3 \cdot - 2}{3 \cdot 1} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})}{3} - \frac{2 \ln (1)}{3}$

Rewrite the expression.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{- 2}{1} (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})}{3} - \frac{2 \ln (1)}{3}$

Divide $- 2$ by $1$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} - 2 (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3})}{3} - \frac{2 \ln (1)}{3}$

Evaluate.

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Combine the numerators over the common denominator.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} - 2 (\frac{2 \cdot 6^{\frac{3}{2}}}{3} - \frac{2}{3}) - 2 \ln (1)}{3}$

Simplify each term.

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Apply the distributive property.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} - 2 \frac{2 \cdot 6^{\frac{3}{2}}}{3} - 2 (- \frac{2}{3}) - 2 \ln (1)}{3}$

Multiply $- 2 \frac{2 \cdot 6^{\frac{3}{2}}}{3}$ .

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Combine $- 2$ and $\frac{2 \cdot 6^{\frac{3}{2}}}{3}$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{- 2 (2 \cdot 6^{\frac{3}{2}})}{3} - 2 (- \frac{2}{3}) - 2 \ln (1)}{3}$

Multiply $2$ by $- 2$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{- 4 \cdot 6^{\frac{3}{2}}}{3} - 2 (- \frac{2}{3}) - 2 \ln (1)}{3}$

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

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Multiply $- 1$ by $- 2$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{- 4 \cdot 6^{\frac{3}{2}}}{3} + 2 (\frac{2}{3}) - 2 \ln (1)}{3}$

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

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{- 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{2 \cdot 2}{3} - 2 \ln (1)}{3}$

Multiply $2$ by $2$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{- 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3} - 2 \ln (1)}{3}$

Move the negative in front of the fraction.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3} - 2 \ln (1)}{3}$

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

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3} - 2 \cdot 0}{3}$

Multiply $- 2$ by $0$ .

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3} + 0}{3}$

Combine the numerators over the common denominator.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} + \frac{- 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Move the negative in front of the fraction.

$\frac{2 \ln (6) \cdot 6^{\frac{3}{2}} - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Reorder factors in $2 \ln (6) \cdot 6^{\frac{3}{2}} - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}$ .

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (6) - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Simplify the numerator.

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Rewrite $\ln (6)$ as $\ln (2 (3))$ .

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (2 (3)) - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

$\frac{2 \cdot 6^{\frac{3}{2}} (\ln (2) + \ln (3)) - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Apply the distributive property.

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (2) + 2 \cdot 6^{\frac{3}{2}} \ln (3) - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

To write $2 \cdot 6^{\frac{3}{2}} \ln (2)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) + 2 \cdot 6^{\frac{3}{2}} \ln (2) \cdot \frac{3}{3} - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Combine $2 \cdot 6^{\frac{3}{2}} \ln (2)$ and $\frac{3}{3}$ .

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) + \frac{2 \cdot 6^{\frac{3}{2}} \ln (2) \cdot 3}{3} - \frac{4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Combine the numerators over the common denominator.

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) + \frac{2 \cdot 6^{\frac{3}{2}} \ln (2) \cdot 3 - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Multiply $2 \cdot 6^{\frac{3}{2}} \ln (2) \cdot 3$ .

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Multiply $3$ by $2$ .

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) + \frac{6 \cdot 6^{\frac{3}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Multiply $6$ by $6^{\frac{3}{2}}$ by adding the exponents.

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Multiply $6$ by $6^{\frac{3}{2}}$ .

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Raise $6$ to the power of $1$ .

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) + \frac{6^{1} \cdot 6^{\frac{3}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

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

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) + \frac{6^{1 + \frac{3}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

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

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) + \frac{6^{\frac{2}{2} + \frac{3}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Combine the numerators over the common denominator.

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) + \frac{6^{\frac{2 + 3}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Add $2$ and $3$ .

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) + \frac{6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

To write $2 \cdot 6^{\frac{3}{2}} \ln (3)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) \cdot \frac{3}{3} + \frac{6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Combine $2 \cdot 6^{\frac{3}{2}} \ln (3)$ and $\frac{3}{3}$ .

$\frac{\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) \cdot 3}{3} + \frac{6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Combine the numerators over the common denominator.

$\frac{\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) \cdot 3 + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}}}{3} + \frac{4}{3}}{3}$

Combine the numerators over the common denominator.

$\frac{\frac{2 \cdot 6^{\frac{3}{2}} \ln (3) \cdot 3 + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3}}{3}$

Reorder factors in $2 \cdot 6^{\frac{3}{2}} \ln (3) \cdot 3 + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4$ .

$\frac{\frac{2 \cdot 3 \cdot 6^{\frac{3}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3}}{3}$

Simplify the numerator.

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Multiply $2$ by $3$ .

$\frac{\frac{6 \cdot 6^{\frac{3}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3}}{3}$

Multiply $6$ by $6^{\frac{3}{2}}$ by adding the exponents.

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Multiply $6$ by $6^{\frac{3}{2}}$ .

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Raise $6$ to the power of $1$ .

$\frac{\frac{6^{1} \cdot 6^{\frac{3}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3}}{3}$

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

$\frac{\frac{6^{1 + \frac{3}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3}}{3}$

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

$\frac{\frac{6^{\frac{2}{2} + \frac{3}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3}}{3}$

Combine the numerators over the common denominator.

$\frac{\frac{6^{\frac{2 + 3}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3}}{3}$

Add $2$ and $3$ .

$\frac{\frac{6^{\frac{5}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3}}{3}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{6^{\frac{5}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3} \cdot \frac{1}{3}$

Multiply $\frac{6^{\frac{5}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3} \cdot \frac{1}{3}$ .

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Multiply $\frac{6^{\frac{5}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3}$ by $\frac{1}{3}$ .

$\frac{6^{\frac{5}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{3 \cdot 3}$

Multiply $3$ by $3$ .

$\frac{6^{\frac{5}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{9}$

The result can be shown in multiple forms.

Exact Form:

$\frac{6^{\frac{5}{2}} \ln (3) + 6^{\frac{5}{2}} \ln (2) - 4 \cdot 6^{\frac{3}{2}} + 4}{9}$

Decimal Form:

$11.46805756 \dots$