Calculus Examples

$\int \frac{1}{\cos^{2} (3 x + 1)} d x d x$

Step 1

Convert from $\frac{1}{\cos^{2} (3 x + 1)}$ to $\sec^{2} (3 x + 1)$ .

$\frac{d}{d x} [\int \sec^{2} (3 x + 1) d x d x]$

Step 2

Since $d$ is constant with respect to $x$ , the derivative of $\int \sec^{2} (3 x + 1) d x d x$ with respect to $x$ is $d \frac{d}{d x} [\int \sec^{2} (3 x + 1) d x x]$ .

$d \frac{d}{d x} [\int \sec^{2} (3 x + 1) d x x]$

Step 3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \int \sec^{2} (3 x + 1) d x$ and $g (x) = x$ .

$d (\int \sec^{2} (3 x + 1) d x \frac{d}{d x} [x] + x \frac{d}{d x} [\int \sec^{2} (3 x + 1) d x])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$d (\int \sec^{2} (3 x + 1) d x \cdot 1 + x \frac{d}{d x} [\int \sec^{2} (3 x + 1) d x])$

Step 4.2

Multiply $\int \sec^{2} (3 x + 1) d x$ by $1$ .

$d (\int \sec^{2} (3 x + 1) d x + x \frac{d}{d x} [\int \sec^{2} (3 x + 1) d x])$

Step 5

$\int \sec^{2} (3 x + 1) d x$ is an antiderivative of $\sec^{2} (3 x + 1)$ , so by definition $\frac{d}{d x} [\int \sec^{2} (3 x + 1) d x]$ is $\sec^{2} (3 x + 1)$ .

$d (\int \sec^{2} (3 x + 1) d x + x \sec^{2} (3 x + 1))$

Step 6

Apply the distributive property.

$d \int \sec^{2} (3 x + 1) d x + d x \sec^{2} (3 x + 1)$