Calculus Examples

$\int_{0}^{\frac{π}{4}} 5 \sec (x) \tan (x) d x$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{0}^{\frac{π}{4}} \sec (x) \tan (x) d x$

Step 2

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$5 (\sec (x)]_{0}^{\frac{π}{4}})$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\sec (x)$ at $\frac{π}{4}$ and at $0$ .

$5 (\sec (\frac{π}{4}) - \sec (0))$

Step 3.2

Simplify.

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Step 3.2.1

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

$5 (\frac{2}{\sqrt{2}} - \sec (0))$

Step 3.2.2

The exact value of $\sec (0)$ is $1$ .

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

Step 3.2.3

Multiply $- 1$ by $1$ .

$5 (\frac{2}{\sqrt{2}} - 1)$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$5 \frac{2}{\sqrt{2}} + 5 \cdot - 1$

Step 3.3.2

Multiply $5 \frac{2}{\sqrt{2}}$ .

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Step 3.3.2.1

Combine $5$ and $\frac{2}{\sqrt{2}}$ .

$\frac{5 \cdot 2}{\sqrt{2}} + 5 \cdot - 1$

Step 3.3.2.2

Multiply $5$ by $2$ .

$\frac{10}{\sqrt{2}} + 5 \cdot - 1$

Step 3.3.3

Multiply $5$ by $- 1$ .

$\frac{10}{\sqrt{2}} - 5$

Step 3.4

Simplify each term.

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Step 3.4.1

Multiply $\frac{10}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{10}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} - 5$

Step 3.4.2

Combine and simplify the denominator.

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Step 3.4.2.1

Multiply $\frac{10}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{10 \sqrt{2}}{\sqrt{2} \sqrt{2}} - 5$

Step 3.4.2.2

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

$\frac{10 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} - 5$

Step 3.4.2.3

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

$\frac{10 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} - 5$

Step 3.4.2.4

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

$\frac{10 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} - 5$

Step 3.4.2.5

Add $1$ and $1$ .

$\frac{10 \sqrt{2}}{{\sqrt{2}}^{2}} - 5$

Step 3.4.2.6

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

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Step 3.4.2.6.1

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

$\frac{10 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} - 5$

Step 3.4.2.6.2

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

$\frac{10 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} - 5$

Step 3.4.2.6.3

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

$\frac{10 \sqrt{2}}{2^{\frac{2}{2}}} - 5$

Step 3.4.2.6.4

Cancel the common factor of $2$ .

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Step 3.4.2.6.4.1

Cancel the common factor.

$\frac{10 \sqrt{2}}{2^{\frac{2}{2}}} - 5$

Step 3.4.2.6.4.2

Rewrite the expression.

$\frac{10 \sqrt{2}}{2^{1}} - 5$

Step 3.4.2.6.5

Evaluate the exponent.

$\frac{10 \sqrt{2}}{2} - 5$

Step 3.4.3

Cancel the common factor of $10$ and $2$ .

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Step 3.4.3.1

Factor $2$ out of $10 \sqrt{2}$ .

$\frac{2 (5 \sqrt{2})}{2} - 5$

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

Factor $2$ out of $2$ .

$\frac{2 (5 \sqrt{2})}{2 (1)} - 5$

Step 3.4.3.2.2

Cancel the common factor.

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

Step 3.4.3.2.3

Rewrite the expression.

$\frac{5 \sqrt{2}}{1} - 5$

Step 3.4.3.2.4

Divide $5 \sqrt{2}$ by $1$ .

$5 \sqrt{2} - 5$

Step 4

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{2} - 5$

Decimal Form:

$2.07106781 \dots$