Calculus Examples

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

Step 1

Let $u = x^{2}$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2}$ .

$\frac{d}{d x} [x^{2}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \cot (u) \csc (u) \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{2}$ and $\cot (u)$ .

$\int \frac{\cot (u)}{2} \csc (u) d u$

Step 2.2

Combine $\frac{\cot (u)}{2}$ and $\csc (u)$ .

$\int \frac{\cot (u) \csc (u)}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \cot (u) \csc (u) d u$

Step 4

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

$\frac{1}{2} (- \csc (u) + C)$

Step 5

Simplify.

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Step 5.1

Simplify.

$\frac{1}{2} (- \csc (u)) + C$

Step 5.2

Combine $\frac{1}{2}$ and $\csc (u)$ .

$- \frac{\csc (u)}{2} + C$

Step 6

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 7

Reorder terms.

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