Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Rewrite using the commutative property of multiplication.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Add $1$ and $1$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

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

Step 5

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

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

Step 6

Simplify.

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