Calculus Examples

$\int_{\frac{π}{6}}^{\frac{π}{4}} \tan (u) d u$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

The integral of $\tan (u)$ with respect to $u$ is $\ln (| \sec (u) |)$ .

$\ln (| \sec (u) |)]_{\frac{π}{6}}^{\frac{π}{4}}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Evaluate $\ln (| \sec (u) |)$ at $\frac{π}{4}$ and at $\frac{π}{6}$ .

$\ln (| \sec (\frac{π}{4}) |) - \ln (| \sec (\frac{π}{6}) |)$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\ln (| \frac{2}{\sqrt{2}} |) - \ln (| \sec (\frac{π}{6}) |)$

Step 3.2.2

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$\ln (| \frac{2}{\sqrt{2}} |) - \ln (| \frac{2}{\sqrt{3}} |)$

Step 3.2.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| \frac{2}{\sqrt{2}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Simplify the numerator.

Tap for more steps...

Step 3.3.1.1

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

$\ln (\frac{| \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2

Combine and simplify the denominator.

Tap for more steps...

Step 3.3.1.2.1

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

$\ln (\frac{| \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$\ln (\frac{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\ln (\frac{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.4

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

$\ln (\frac{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.5

Add $1$ and $1$ .

$\ln (\frac{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 3.3.1.2.6.1

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

$\ln (\frac{| \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.6.2

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

$\ln (\frac{| \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.6.3

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

$\ln (\frac{| \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.2.6.4.1

Cancel the common factor.

$\ln (\frac{| \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.6.4.2

Rewrite the expression.

$\ln (\frac{| \frac{2 \sqrt{2}}{2^{1}} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.2.6.5

Evaluate the exponent.

$\ln (\frac{| \frac{2 \sqrt{2}}{2} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.3.1

Cancel the common factor.

$\ln (\frac{| \frac{2 \sqrt{2}}{2} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.3.2

Divide $\sqrt{2}$ by $1$ .

$\ln (\frac{| \sqrt{2} |}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.1.4

$\sqrt{2}$ is approximately $1.41421356$ which is positive so remove the absolute value

$\ln (\frac{\sqrt{2}}{| \frac{2}{\sqrt{3}} |})$

Step 3.3.2

Simplify the denominator.

Tap for more steps...

Step 3.3.2.1

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

$\ln (\frac{\sqrt{2}}{| \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} |})$

Step 3.3.2.2

Combine and simplify the denominator.

Tap for more steps...

Step 3.3.2.2.1

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

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} |})$

Step 3.3.2.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} |})$

Step 3.3.2.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} |})$

Step 3.3.2.2.4

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

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} |})$

Step 3.3.2.2.5

Add $1$ and $1$ .

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} |})$

Step 3.3.2.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 3.3.2.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} |})$

Step 3.3.2.2.6.2

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

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} |})$

Step 3.3.2.2.6.3

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

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} |})$

Step 3.3.2.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.2.6.4.1

Cancel the common factor.

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} |})$

Step 3.3.2.2.6.4.2

Rewrite the expression.

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{3^{1}} |})$

Step 3.3.2.2.6.5

Evaluate the exponent.

$\ln (\frac{\sqrt{2}}{| \frac{2 \sqrt{3}}{3} |})$

Step 3.3.2.3

$\frac{2 \sqrt{3}}{3}$ is approximately $1.15470053$ which is positive so remove the absolute value

$\ln (\frac{\sqrt{2}}{\frac{2 \sqrt{3}}{3}})$

Step 3.3.3

Multiply the numerator by the reciprocal of the denominator.

$\ln (\sqrt{2} \frac{3}{2 \sqrt{3}})$

Step 3.3.4

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

$\ln (\frac{\sqrt{2} \cdot 3}{2 \sqrt{3}})$

Step 3.3.5

Move $3$ to the left of $\sqrt{2}$ .

$\ln (\frac{3 \sqrt{2}}{2 \sqrt{3}})$

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

$\ln (\frac{3 \sqrt{2}}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 4.2

Combine and simplify the denominator.

Tap for more steps...

Step 4.2.1

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

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 \sqrt{3} \sqrt{3}})$

Step 4.2.2

Move $\sqrt{3}$ .

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 (\sqrt{3} \sqrt{3})})$

Step 4.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 ({\sqrt{3}}^{1} \sqrt{3})})$

Step 4.2.4

Raise $\sqrt{3}$ to the power of $1$ .

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})})$

Step 4.2.5

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

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 {\sqrt{3}}^{1 + 1}})$

Step 4.2.6

Add $1$ and $1$ .

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 {\sqrt{3}}^{2}})$

Step 4.2.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 4.2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 {(3^{\frac{1}{2}})}^{2}})$

Step 4.2.7.2

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

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

Step 4.2.7.3

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

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}})$

Step 4.2.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.7.4.1

Cancel the common factor.

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}})$

Step 4.2.7.4.2

Rewrite the expression.

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

Step 4.2.7.5

Evaluate the exponent.

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 \cdot 3})$

Step 4.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.1

Cancel the common factor.

$\ln (\frac{3 \sqrt{2} \sqrt{3}}{2 \cdot 3})$

Step 4.3.2

Rewrite the expression.

$\ln (\frac{\sqrt{2} \sqrt{3}}{2})$

Step 4.4

Simplify the numerator.

Tap for more steps...

Step 4.4.1

Combine using the product rule for radicals.

$\ln (\frac{\sqrt{2 \cdot 3}}{2})$

Step 4.4.2

Multiply $2$ by $3$ .

$\ln (\frac{\sqrt{6}}{2})$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{\sqrt{6}}{2})$

Decimal Form:

$0.20273255 \dots$