Calculus Examples

$\int \sec (x) (5 \sec (x) + 3 \tan (x)) d x$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Apply the distributive property.

$\int \sec (x) (5 \sec (x)) + \sec (x) (3 \tan (x)) d x$

Step 1.2

Rewrite using the commutative property of multiplication.

$\int 5 \sec (x) \sec (x) + \sec (x) (3 \tan (x)) d x$

Step 1.3

Rewrite using the commutative property of multiplication.

$\int 5 \sec (x) \sec (x) + 3 \sec (x) \tan (x) d x$

Step 1.4

Multiply $5 \sec (x) \sec (x)$ .

Tap for more steps...

Step 1.4.1

Raise $\sec (x)$ to the power of $1$ .

$\int 5 (\sec^{1} (x) \sec (x)) + 3 \sec (x) \tan (x) d x$

Step 1.4.2

Raise $\sec (x)$ to the power of $1$ .

$\int 5 (\sec^{1} (x) \sec^{1} (x)) + 3 \sec (x) \tan (x) d x$

Step 1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 5 {\sec (x)}^{1 + 1} + 3 \sec (x) \tan (x) d x$

Step 1.4.4

Add $1$ and $1$ .

$\int 5 \sec^{2} (x) + 3 \sec (x) \tan (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 5 \sec^{2} (x) d x + \int 3 \sec (x) \tan (x) d x$

Step 3

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \sec^{2} (x) d x + \int 3 \sec (x) \tan (x) d x$

Step 4

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

$5 (\tan (x) + C) + \int 3 \sec (x) \tan (x) d x$

Step 5

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

$5 (\tan (x) + C) + 3 \int \sec (x) \tan (x) d x$

Step 6

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

$5 (\tan (x) + C) + 3 (\sec (x) + C)$

Step 7

Simplify.

$5 \tan (x) + 3 \sec (x) + C$