Calculus Examples

$\int_{0}^{1} \tan^{2} (\frac{π}{4} y) d y$

Step 1

Let $u = \frac{π}{4} y$ . Then $d u = \frac{π}{4} d y$ , so $\frac{4}{π} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{π}{4} y$ . Find $\frac{d u}{d y}$ .

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Step 1.1.1

Differentiate $\frac{π}{4} y$ .

$\frac{d}{d y} [\frac{π}{4} y]$

Step 1.1.2

Since $\frac{π}{4}$ is constant with respect to $y$ , the derivative of $\frac{π}{4} y$ with respect to $y$ is $\frac{π}{4} \frac{d}{d y} [y]$ .

$\frac{π}{4} \frac{d}{d y} [y]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$\frac{π}{4} \cdot 1$

Step 1.1.4

Multiply $\frac{π}{4}$ by $1$ .

$\frac{π}{4}$

Step 1.2

Substitute the lower limit in for $y$ in $u = \frac{π}{4} y$ .

$u_{lower} = \frac{π}{4} \cdot 0$

Step 1.3

Multiply $\frac{π}{4}$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $y$ in $u = \frac{π}{4} y$ .

$u_{upper} = \frac{π}{4} \cdot 1$

Step 1.5

Multiply $\frac{π}{4}$ by $1$ .

$u_{upper} = \frac{π}{4}$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{4}$

Step 1.7

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

$\int_{0}^{\frac{π}{4}} \tan^{2} (u) \frac{1}{\frac{π}{4}} d u$

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{π}{4}$ .

$\int_{0}^{\frac{π}{4}} \tan^{2} (u) (1 \frac{4}{π}) d u$

Step 2.2

Multiply $\frac{4}{π}$ by $1$ .

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

Step 2.3

Combine $\tan^{2} (u)$ and $\frac{4}{π}$ .

$\int_{0}^{\frac{π}{4}} \frac{\tan^{2} (u) \cdot 4}{π} d u$

Step 2.4

Move $4$ to the left of $\tan^{2} (u)$ .

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

Step 3

Since $\frac{4}{π}$ is constant with respect to $u$ , move $\frac{4}{π}$ out of the integral.

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

Step 4

Using the Pythagorean Identity, rewrite $\tan^{2} (u)$ as $- 1 + \sec^{2} (u)$ .

$\frac{4}{π} \int_{0}^{\frac{π}{4}} - 1 + \sec^{2} (u) d u$

Step 5

Split the single integral into multiple integrals.

$\frac{4}{π} (\int_{0}^{\frac{π}{4}} - 1 d u + \int_{0}^{\frac{π}{4}} \sec^{2} (u) d u)$

Step 6

Apply the constant rule.

$\frac{4}{π} (- u]_{0}^{\frac{π}{4}} + \int_{0}^{\frac{π}{4}} \sec^{2} (u) d u)$

Step 7

Since the derivative of $\tan (u)$ is $\sec^{2} (u)$ , the integral of $\sec^{2} (u)$ is $\tan (u)$ .

$\frac{4}{π} (- u]_{0}^{\frac{π}{4}} + \tan (u)]_{0}^{\frac{π}{4}})$

Step 8

Combine $- u]_{0}^{\frac{π}{4}}$ and $\tan (u)]_{0}^{\frac{π}{4}}$ .

$\frac{4}{π} - u + \tan (u)]_{0}^{\frac{π}{4}}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $- u + \tan (u)$ at $\frac{π}{4}$ and at $0$ .

$\frac{4}{π} ((- \frac{π}{4} + \tan (\frac{π}{4})) - (- 0 + \tan (0)))$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $- 1$ by $0$ .

$\frac{4}{π} (- \frac{π}{4} + \tan (\frac{π}{4}) - (0 + \tan (0)))$

Step 9.2.2

Add $0$ and $\tan (0)$ .

$\frac{4}{π} (- \frac{π}{4} + \tan (\frac{π}{4}) - \tan (0))$

Step 10

Simplify.

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Step 10.1

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{4}{π} (- \frac{π}{4} + 1 - \tan (0))$

Step 10.2

The exact value of $\tan (0)$ is $0$ .

$\frac{4}{π} (- \frac{π}{4} + 1 - 0)$

Step 10.3

Multiply $- 1$ by $0$ .

$\frac{4}{π} (- \frac{π}{4} + 1 + 0)$

Step 10.4

Add $- \frac{π}{4} + 1$ and $0$ .

$\frac{4}{π} (- \frac{π}{4} + 1)$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\frac{4}{π} (- \frac{π}{4}) + \frac{4}{π} \cdot 1$

Step 11.2

Cancel the common factor of $4$ .

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Step 11.2.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

$\frac{4}{π} \cdot \frac{- π}{4} + \frac{4}{π} \cdot 1$

Step 11.2.2

Cancel the common factor.

$\frac{4}{π} \cdot \frac{- π}{4} + \frac{4}{π} \cdot 1$

Step 11.2.3

Rewrite the expression.

$\frac{1}{π} (- π) + \frac{4}{π} \cdot 1$

Step 11.3

Cancel the common factor of $π$ .

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Step 11.3.1

Factor $π$ out of $- π$ .

$\frac{1}{π} (π \cdot - 1) + \frac{4}{π} \cdot 1$

Step 11.3.2

Cancel the common factor.

$\frac{1}{π} (π \cdot - 1) + \frac{4}{π} \cdot 1$

Step 11.3.3

Rewrite the expression.

$- 1 + \frac{4}{π} \cdot 1$

Step 11.4

Multiply $\frac{4}{π}$ by $1$ .

$- 1 + \frac{4}{π}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- 1 + \frac{4}{π}$

Decimal Form:

$0.27323954 \dots$