Calculus Examples

$\int \sec (3 t) \tan (3 t) d t$

Step 1

Let $u_{2} = \sec (3 t)$ . Then $d u_{2} = 3 \sec (3 t) \tan (3 t) d t$ , so $\frac{1}{3} d u_{2} = \sec (3 t) \tan (3 t) d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \sec (3 t)$ . Find $\frac{d u_{2}}{d t}$ .

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Step 1.1.1

Differentiate $\sec (3 t)$ .

$\frac{d}{d t} [\sec (3 t)]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \sec (t)$ and $g (t) = 3 t$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $3 t$ .

$\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d t} [3 t]$

Step 1.1.2.2

The derivative of $\sec (u_{1})$ with respect to $u_{1}$ is $\sec (u_{1}) \tan (u_{1})$ .

$\sec (u_{1}) \tan (u_{1}) \frac{d}{d t} [3 t]$

Step 1.1.2.3

Replace all occurrences of $u_{1}$ with $3 t$ .

$\sec (3 t) \tan (3 t) \frac{d}{d t} [3 t]$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Since $3$ is constant with respect to $t$ , the derivative of $3 t$ with respect to $t$ is $3 \frac{d}{d t} [t]$ .

$\sec (3 t) \tan (3 t) (3 \frac{d}{d t} [t])$

Step 1.1.3.2

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

$\sec (3 t) \tan (3 t) (3 \cdot 1)$

Step 1.1.3.3

Simplify the expression.

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Step 1.1.3.3.1

Multiply $3$ by $1$ .

$\sec (3 t) \tan (3 t) \cdot 3$

Step 1.1.3.3.2

Move $3$ to the left of $\sec (3 t) \tan (3 t)$ .

$3 \sec (3 t) \tan (3 t)$

Step 1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int \frac{1}{3} d u_{2}$

Step 2

Apply the constant rule.

$\frac{1}{3} u_{2} + C$

Step 3

Replace all occurrences of $u_{2}$ with $\sec (3 t)$ .

$\frac{1}{3} \sec (3 t) + C$