Calculus Examples

$\int 2 \sec (x) \tan (x) + 7 \sec^{2} (x) d x$

Step 1

Split the single integral into multiple integrals.

$\int 2 \sec (x) \tan (x) d x + \int 7 \sec^{2} (x) d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \sec (x) \tan (x) d x + \int 7 \sec^{2} (x) d x$

Step 3

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

$2 (\sec (x) + C) + \int 7 \sec^{2} (x) d x$

Step 4

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$2 (\sec (x) + C) + 7 \int \sec^{2} (x) d x$

Step 5

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

$2 (\sec (x) + C) + 7 (\tan (x) + C)$

Step 6

Simplify.

$2 \sec (x) + 7 \tan (x) + C$