Calculus Examples

$\frac{d y}{d x} = \csc (x) (\csc (x) - \cot (x))$

Step 1

Rewrite the equation.

$d y = \csc (x) (\csc (x) - \cot (x)) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \csc (x) (\csc (x) - \cot (x)) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \csc (x) (\csc (x) - \cot (x)) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Simplify.

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Step 2.3.1.1

Apply the distributive property.

$y + C_{1} = \int \csc (x) \csc (x) + \csc (x) (- \cot (x)) d x$

Step 2.3.1.2

Multiply $\csc (x) \csc (x)$ .

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Step 2.3.1.2.1

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

$y + C_{1} = \int \csc^{1} (x) \csc (x) + \csc (x) (- \cot (x)) d x$

Step 2.3.1.2.2

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

$y + C_{1} = \int \csc^{1} (x) \csc^{1} (x) + \csc (x) (- \cot (x)) d x$

Step 2.3.1.2.3

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

$y + C_{1} = \int {\csc (x)}^{1 + 1} + \csc (x) (- \cot (x)) d x$

Step 2.3.1.2.4

Add $1$ and $1$ .

$y + C_{1} = \int \csc^{2} (x) + \csc (x) (- \cot (x)) d x$

Step 2.3.1.3

Rewrite using the commutative property of multiplication.

$y + C_{1} = \int \csc^{2} (x) - \csc (x) \cot (x) d x$

Step 2.3.2

Split the single integral into multiple integrals.

$y + C_{1} = \int \csc^{2} (x) d x + \int - \csc (x) \cot (x) d x$

Step 2.3.3

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

$y + C_{1} = - \cot (x) + C_{2} + \int - \csc (x) \cot (x) d x$

Step 2.3.4

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

$y + C_{1} = - \cot (x) + C_{2} - \int \csc (x) \cot (x) d x$

Step 2.3.5

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

$y + C_{1} = - \cot (x) + C_{2} - (- \csc (x) + C_{3})$

Step 2.3.6

Simplify.

$y + C_{1} = - \cot (x) + \csc (x) + C_{4}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = - \cot (x) + \csc (x) + K$