Calculus Examples

$\int 8 \csc (y) (\cot (y) - \csc (y)) d y$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int 8 \csc (y) \cot (y) + 8 \csc (y) (- \csc (y)) d y$

Step 1.2

Multiply $8 \csc (y) (- \csc (y))$ .

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Step 1.2.1

Multiply $- 1$ by $8$ .

$\int 8 \csc (y) \cot (y) - 8 \csc (y) \csc (y) d y$

Step 1.2.2

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

$\int 8 \csc (y) \cot (y) - 8 (\csc^{1} (y) \csc (y)) d y$

Step 1.2.3

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

$\int 8 \csc (y) \cot (y) - 8 (\csc^{1} (y) \csc^{1} (y)) d y$

Step 1.2.4

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

$\int 8 \csc (y) \cot (y) - 8 {\csc (y)}^{1 + 1} d y$

Step 1.2.5

Add $1$ and $1$ .

$\int 8 \csc (y) \cot (y) - 8 \csc^{2} (y) d y$

Step 2

Split the single integral into multiple integrals.

$\int 8 \csc (y) \cot (y) d y + \int - 8 \csc^{2} (y) d y$

Step 3

Since $8$ is constant with respect to $y$ , move $8$ out of the integral.

$8 \int \csc (y) \cot (y) d y + \int - 8 \csc^{2} (y) d y$

Step 4

Since the derivative of $- \csc (y)$ is $\csc (y) \cot (y)$ , the integral of $\csc (y) \cot (y)$ is $- \csc (y)$ .

$8 (- \csc (y) + C) + \int - 8 \csc^{2} (y) d y$

Step 5

Since $- 8$ is constant with respect to $y$ , move $- 8$ out of the integral.

$8 (- \csc (y) + C) - 8 \int \csc^{2} (y) d y$

Step 6

Since the derivative of $- \cot (y)$ is $\csc^{2} (y)$ , the integral of $\csc^{2} (y)$ is $- \cot (y)$ .

$8 (- \csc (y) + C) - 8 (- \cot (y) + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$- 8 \csc (y) - 8 (- \cot (y)) + C$

Step 7.2

Multiply $- 1$ by $- 8$ .

$- 8 \csc (y) + 8 \cot (y) + C$