Calculus Examples

$\int (4 x - \csc^{2} (x)) d x$

Step 1

Remove parentheses.

$\int 4 x - \csc^{2} (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 4 x d x + \int - \csc^{2} (x) d x$

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int x d x + \int - \csc^{2} (x) d x$

Step 4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$4 (\frac{1}{2} x^{2} + C) + \int - \csc^{2} (x) d x$

Step 5

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

$4 (\frac{1}{2} x^{2} + C) - \int \csc^{2} (x) d x$

Step 6

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

$4 (\frac{1}{2} x^{2} + C) - (- \cot (x) + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$4 (\frac{1}{2}) x^{2} - - \cot (x) + C$

Step 7.2

Simplify.

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Step 7.2.1

Combine $4$ and $\frac{1}{2}$ .

$\frac{4}{2} x^{2} - - \cot (x) + C$

Step 7.2.2

Cancel the common factor of $4$ and $2$ .

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Step 7.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} x^{2} - - \cot (x) + C$

Step 7.2.2.2

Cancel the common factors.

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Step 7.2.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} x^{2} - - \cot (x) + C$

Step 7.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} x^{2} - - \cot (x) + C$

Step 7.2.2.2.3

Rewrite the expression.

$\frac{2}{1} x^{2} - - \cot (x) + C$

Step 7.2.2.2.4

Divide $2$ by $1$ .

$2 x^{2} - - \cot (x) + C$

Step 7.2.3

Multiply $- 1$ by $- 1$ .

$2 x^{2} + 1 \cot (x) + C$

Step 7.2.4

Multiply $\cot (x)$ by $1$ .

$2 x^{2} + \cot (x) + C$