Calculus Examples

$\int \sec^{2} (4 x + 1) d x$

Step 1

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

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Step 1.1

Let $u = 4 x + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x + 1$ .

$\frac{d}{d x} [4 x + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $4 x + 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.1.3.1

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

$4 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

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 $n = 1$ .

$4 \cdot 1 + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$4 + 0$

Step 1.1.4.2

Add $4$ and $0$ .

$4$

Step 1.2

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

$\int \sec^{2} (u) \frac{1}{4} d u$

Step 2

Combine $\sec^{2} (u)$ and $\frac{1}{4}$ .

$\int \frac{\sec^{2} (u)}{4} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \sec^{2} (u) d u$

Step 4

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

$\frac{1}{4} (\tan (u) + C)$

Step 5

Simplify.

$\frac{1}{4} \tan (u) + C$

Step 6

Replace all occurrences of $u$ with $4 x + 1$ .

$\frac{1}{4} \tan (4 x + 1) + C$