Calculus Examples

$\int 2 \sec (y) (\tan (y) - \sec (y)) d y$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int 2 \sec (y) \tan (y) + 2 \sec (y) (- \sec (y)) d y$

Step 1.2

Multiply $2 \sec (y) (- \sec (y))$ .

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Step 1.2.1

Multiply $- 1$ by $2$ .

$\int 2 \sec (y) \tan (y) - 2 \sec (y) \sec (y) d y$

Step 1.2.2

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

$\int 2 \sec (y) \tan (y) - 2 (\sec^{1} (y) \sec (y)) d y$

Step 1.2.3

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

$\int 2 \sec (y) \tan (y) - 2 (\sec^{1} (y) \sec^{1} (y)) d y$

Step 1.2.4

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

$\int 2 \sec (y) \tan (y) - 2 {\sec (y)}^{1 + 1} d y$

Step 1.2.5

Add $1$ and $1$ .

$\int 2 \sec (y) \tan (y) - 2 \sec^{2} (y) d y$

Step 2

Split the single integral into multiple integrals.

$\int 2 \sec (y) \tan (y) d y + \int - 2 \sec^{2} (y) d y$

Step 3

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

$2 \int \sec (y) \tan (y) d y + \int - 2 \sec^{2} (y) d y$

Step 4

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

$2 (\sec (y) + C) + \int - 2 \sec^{2} (y) d y$

Step 5

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

$2 (\sec (y) + C) - 2 \int \sec^{2} (y) d y$

Step 6

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

$2 (\sec (y) + C) - 2 (\tan (y) + C)$

Step 7

Simplify.

$2 \sec (y) - 2 \tan (y) + C$