Calculus Examples

$\int 3 \sqrt{\tan^{2} (4 x) + 1} d x$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Apply pythagorean identity.

$\int 3 \sqrt{\sec^{2} (4 x)} d x$

Step 1.2

Pull terms out from under the radical, assuming positive real numbers.

$\int 3 \sec (4 x) d x$

Step 2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \sec (4 x) d x$

Step 3

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

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Differentiate $4 x$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

$4 \cdot 1$

Step 3.1.4

Multiply $4$ by $1$ .

$4$

Step 3.2

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

$3 \int \sec (u) \frac{1}{4} d u$

Step 4

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

$3 \int \frac{\sec (u)}{4} d u$

Step 5

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

$3 (\frac{1}{4} \int \sec (u) d u)$

Step 6

Combine $\frac{1}{4}$ and $3$ .

$\frac{3}{4} \int \sec (u) d u$

Step 7

The integral of $\sec (u)$ with respect to $u$ is $\ln (| \sec (u) + \tan (u) |)$ .

$\frac{3}{4} (\ln (| \sec (u) + \tan (u) |) + C)$

Step 8

Simplify.

$\frac{3}{4} \ln (| \sec (u) + \tan (u) |) + C$

Step 9

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

$\frac{3}{4} \ln (| \sec (4 x) + \tan (4 x) |) + C$