Calculus Examples

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

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\sec (x)$ .

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

Step 1.1.2

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

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

Step 1.2

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

$u_{lower} = \sec (- \frac{π}{4})$

Step 1.3

Simplify.

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Step 1.3.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$u_{lower} = \sec (\frac{7 π}{4})$

Step 1.3.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$u_{lower} = \sec (\frac{π}{4})$

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Combine and simplify the denominator.

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Step 1.3.5.1

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

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

Step 1.3.5.2

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

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

Step 1.3.5.3

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

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

Step 1.3.5.4

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

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

Step 1.3.5.5

Add $1$ and $1$ .

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

Step 1.3.5.6

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

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Step 1.3.5.6.1

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

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

Step 1.3.5.6.2

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

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

Step 1.3.5.6.3

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

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

Step 1.3.5.6.4

Cancel the common factor of $2$ .

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Step 1.3.5.6.4.1

Cancel the common factor.

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

Step 1.3.5.6.4.2

Rewrite the expression.

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

Step 1.3.5.6.5

Evaluate the exponent.

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

Step 1.3.6

Cancel the common factor of $2$ .

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Step 1.3.6.1

Cancel the common factor.

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

Step 1.3.6.2

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

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

Step 1.4

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

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

Step 1.5

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

$u_{upper} = 1$

Step 1.6

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

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

$u_{upper} = 1$

Step 1.7

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

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

Step 2

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

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

Step 3

Simplify by cancelling exponent with radical.

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Step 3.1

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

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

Step 3.2

Simplify the expression.

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Step 3.2.1

One to any power is one.

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

Step 3.2.2

Multiply $\frac{1}{2}$ by $1$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Cancel the common factor of $2$ .

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Step 3.3.4.1

Cancel the common factor.

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

Step 3.3.4.2

Rewrite the expression.

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

Step 3.3.5

Evaluate the exponent.

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

Step 3.4

Simplify the expression.

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Step 3.4.1

Simplify.

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Step 3.4.1.1

Multiply $2$ by $- 1$ .

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

Step 3.4.1.2

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

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

Step 3.4.1.3

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

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Step 3.4.1.3.1

Factor $2$ out of $- 2$ .

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

Step 3.4.1.3.2

Cancel the common factors.

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Step 3.4.1.3.2.1

Factor $2$ out of $2$ .

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

Step 3.4.1.3.2.2

Cancel the common factor.

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

Step 3.4.1.3.2.3

Rewrite the expression.

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

Step 3.4.1.3.2.4

Divide $- 1$ by $1$ .

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

Step 3.4.2

Simplify.

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Step 3.4.2.1

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

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

Step 3.4.2.2

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

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

Step 3.4.2.3

Combine the numerators over the common denominator.

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

Step 3.4.2.4

Simplify the numerator.

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Step 3.4.2.4.1

Multiply $- 1$ by $2$ .

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

Step 3.4.2.4.2

Subtract $2$ from $1$ .

$\frac{- 1}{2}$

Step 3.4.2.5

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$