Calculus Examples

$\int 20 x^{3} \sec (5 x^{4} - 1) \tan (5 x^{4} - 1) d x$

Step 1

Since $20$ is constant with respect to $x$ , move $20$ out of the integral.

$20 \int x^{3} \sec (5 x^{4} - 1) \tan (5 x^{4} - 1) d x$

Step 2

Let $u = 5 x^{4} - 1$ . Then $d u = 20 x^{3} d x$ , so $\frac{1}{20} d u = x^{3} d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.1

Let $u = 5 x^{4} - 1$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $5 x^{4} - 1$ .

$\frac{d}{d x} [5 x^{4} - 1]$

Step 2.1.2

By the Sum Rule, the derivative of $5 x^{4} - 1$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 1]$

Step 2.1.3

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

Tap for more steps...

Step 2.1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 1]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$5 (4 x^{3}) + \frac{d}{d x} [- 1]$

Step 2.1.3.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [- 1]$

Step 2.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.4.1

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

$20 x^{3} + 0$

Step 2.1.4.2

Add $20 x^{3}$ and $0$ .

$20 x^{3}$

Step 2.2

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

$20 \int \sec (u) \tan (u) \frac{1}{20} d u$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Combine $\frac{1}{20}$ and $\sec (u)$ .

$20 \int \frac{\sec (u)}{20} \tan (u) d u$

Step 3.2

Combine $\frac{\sec (u)}{20}$ and $\tan (u)$ .

$20 \int \frac{\sec (u) \tan (u)}{20} d u$

Step 4

Since $\frac{1}{20}$ is constant with respect to $u$ , move $\frac{1}{20}$ out of the integral.

$20 (\frac{1}{20} \int \sec (u) \tan (u) d u)$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Combine $\frac{1}{20}$ and $20$ .

$\frac{20}{20} \int \sec (u) \tan (u) d u$

Step 5.2

Cancel the common factor of $20$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor.

$\frac{20}{20} \int \sec (u) \tan (u) d u$

Step 5.2.2

Rewrite the expression.

$1 \int \sec (u) \tan (u) d u$

Step 5.3

Multiply $\int \sec (u) \tan (u) d u$ by $1$ .

$\int \sec (u) \tan (u) d u$

Step 6

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

$\sec (u) + C$

Step 7

Replace all occurrences of $u$ with $5 x^{4} - 1$ .

$\sec (5 x^{4} - 1) + C$