Calculus Examples

$\int_{0}^{\frac{π}{4}} 2 \tan (x) \sec^{2} (x) d x$

Step 1

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

$2 \int_{0}^{\frac{π}{4}} \tan (x) \sec^{2} (x) d x$

Step 2

Let $u = \sec (x)$ . Then $d u = \sec (x) \tan (x) d x$ , so $\frac{1}{\sec (x) \tan (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = \sec (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\sec (x)$ .

$\frac{d}{d x} [\sec (x)]$

Step 2.1.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sec (x) \tan (x)$

Step 2.2

Substitute the lower limit in for $x$ in $u = \sec (x)$ .

$u_{lower} = \sec (0)$

Step 2.3

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

$u_{lower} = 1$

Step 2.4

Substitute the upper limit in for $x$ in $u = \sec (x)$ .

$u_{upper} = \sec (\frac{π}{4})$

Step 2.5

Simplify.

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Step 2.5.1

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

$u_{upper} = \frac{2}{\sqrt{2}}$

Step 2.5.2

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

$u_{upper} = \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.5.3

Combine and simplify the denominator.

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Step 2.5.3.1

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

$u_{upper} = \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.5.3.2

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

$u_{upper} = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.5.3.3

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

$u_{upper} = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.5.3.4

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

$u_{upper} = \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.5.3.5

Add $1$ and $1$ .

$u_{upper} = \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.5.3.6

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

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Step 2.5.3.6.1

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

$u_{upper} = \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.5.3.6.2

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

$u_{upper} = \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.5.3.6.3

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

$u_{upper} = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.5.3.6.4

Cancel the common factor of $2$ .

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Step 2.5.3.6.4.1

Cancel the common factor.

$u_{upper} = \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.5.3.6.4.2

Rewrite the expression.

$u_{upper} = \frac{2 \sqrt{2}}{2^{1}}$

Step 2.5.3.6.5

Evaluate the exponent.

$u_{upper} = \frac{2 \sqrt{2}}{2}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$u_{upper} = \frac{2 \sqrt{2}}{2}$

Step 2.5.4.2

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

$u_{upper} = \sqrt{2}$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = \sqrt{2}$

Step 2.7

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

$2 \int_{1}^{\sqrt{2}} u d u$

Step 3

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

$2 (\frac{1}{2} u^{2}]_{1}^{\sqrt{2}})$

Step 4

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

$2 (\frac{u^{2}}{2}]_{1}^{\sqrt{2}})$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{u^{2}}{2}$ at $\sqrt{2}$ and at $1$ .

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

Step 5.2

Simplify.

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Step 5.2.1

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

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Step 5.2.1.1

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

$2 (\frac{{(2^{\frac{1}{2}})}^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.1.2

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

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

Step 5.2.1.3

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

$2 (\frac{2^{\frac{2}{2}}}{2} - \frac{1^{2}}{2})$

Step 5.2.1.4

Cancel the common factor of $2$ .

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Step 5.2.1.4.1

Cancel the common factor.

$2 (\frac{2^{\frac{2}{2}}}{2} - \frac{1^{2}}{2})$

Step 5.2.1.4.2

Rewrite the expression.

$2 (\frac{2^{1}}{2} - \frac{1^{2}}{2})$

Step 5.2.1.5

Evaluate the exponent.

$2 (\frac{2}{2} - \frac{1^{2}}{2})$

Step 5.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.1

Cancel the common factor.

$2 (\frac{2}{2} - \frac{1^{2}}{2})$

Step 5.2.2.2

Rewrite the expression.

$2 (1 - \frac{1^{2}}{2})$

Step 5.2.3

One to any power is one.

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

Step 5.2.4

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

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

Step 5.2.5

Combine the numerators over the common denominator.

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

Step 5.2.6

Subtract $1$ from $2$ .

$2 (\frac{1}{2})$

Step 5.2.7

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

$\frac{2}{2}$

Step 5.2.8

Cancel the common factor of $2$ .

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Step 5.2.8.1

Cancel the common factor.

$\frac{2}{2}$

Step 5.2.8.2

Rewrite the expression.

$1$