Calculus Examples

$\int \sec (3 x + 5) \tan (3 x + 5) d x$

Step 1

Let $u = 3 x + 5$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 x + 5$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 x + 5$ .

$\frac{d}{d x} [3 x + 5]$

Step 1.1.2

By the Sum Rule, the derivative of $3 x + 5$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [5]$

Step 1.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.1.3.1

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.1.3.2

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

$3 \cdot 1 + \frac{d}{d x} [5]$

Step 1.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [5]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$3 + 0$

Step 1.1.4.2

Add $3$ and $0$ .

$3$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \sec (u) \tan (u) \frac{1}{3} d u$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{3}$ and $\sec (u)$ .

$\int \frac{\sec (u)}{3} \tan (u) d u$

Step 2.2

Combine $\frac{\sec (u)}{3}$ and $\tan (u)$ .

$\int \frac{\sec (u) \tan (u)}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \sec (u) \tan (u) d u$

Step 4

Since the derivative of $\sec (u)$ is $\sec (u) \tan (u)$ , the integral of $\sec (u) \tan (u)$ is $\sec (u)$ .

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

Step 5

Simplify.

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

Step 6

Replace all occurrences of $u$ with $3 x + 5$ .

$\frac{1}{3} \sec (3 x + 5) + C$