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.

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