Calculus Examples

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

Step 1

Let $u = \frac{t}{4}$ . Then $d u = \frac{1}{4} d t$ , so $4 d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $\frac{t}{4}$ .

$\frac{d}{d t} [\frac{t}{4}]$

Step 1.1.2

Since $\frac{1}{4}$ is constant with respect to $t$ , the derivative of $\frac{t}{4}$ with respect to $t$ is $\frac{1}{4} \frac{d}{d t} [t]$ .

$\frac{1}{4} \frac{d}{d t} [t]$

Step 1.1.3

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

$\frac{1}{4} \cdot 1$

Step 1.1.4

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4}$

Step 1.2

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

$u_{lower} = \frac{0}{4}$

Step 1.3

Divide $0$ by $4$ .

$u_{lower} = 0$

Step 1.4

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

$u_{upper} = \frac{π}{4}$

Step 1.5

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 1.6

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

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

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{4}$ .

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

Step 2.2

Multiply $4$ by $1$ .

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

Step 2.3

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

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

Step 3

Since $4$ is constant with respect to $u$ , move $4$ out of the integral.

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

Step 4

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

$4 (\tan (u)]_{0}^{\frac{π}{4}})$

Step 5

Simplify.

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Step 5.1

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

$4 ((\tan (\frac{π}{4})) - \tan (0))$

Step 5.2

Simplify.

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Step 5.2.1

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

$4 (1 - \tan (0))$

Step 5.2.2

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

$4 (1 - 0)$

Step 5.2.3

Multiply $- 1$ by $0$ .

$4 (1 + 0)$

Step 5.2.4

Add $1$ and $0$ .

$4 \cdot 1$

Step 5.2.5

Multiply $4$ by $1$ .

$4$