Calculus Examples

$\int [12 x^{2} + \csc^{2} (3 x)] d x$

Step 1

Remove parentheses.

$\int 12 x^{2} + \csc^{2} (3 x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 12 x^{2} d x + \int \csc^{2} (3 x) d x$

Step 3

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int x^{2} d x + \int \csc^{2} (3 x) d x$

Step 4

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$12 (\frac{1}{3} x^{3} + C) + \int \csc^{2} (3 x) d x$

Step 5

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 5.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 5.1.3

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

$3 \cdot 1$

Step 5.1.4

Multiply $3$ by $1$ .

$3$

Step 5.2

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

$12 (\frac{1}{3} x^{3} + C) + \int \csc^{2} (u) \frac{1}{3} d u$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$12 (\frac{x^{3}}{3} + C) + \int \csc^{2} (u) \frac{1}{3} d u$

Step 6.2

Combine $\csc^{2} (u)$ and $\frac{1}{3}$ .

$12 (\frac{x^{3}}{3} + C) + \int \frac{\csc^{2} (u)}{3} d u$

Step 7

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$12 (\frac{x^{3}}{3} + C) + \frac{1}{3} \int \csc^{2} (u) d u$

Step 8

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

$12 (\frac{x^{3}}{3} + C) + \frac{1}{3} (- \cot (u) + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$4 x^{3} + \frac{1}{3} (- \cot (u)) + C$

Step 9.2

Combine $\frac{1}{3}$ and $\cot (u)$ .

$4 x^{3} - \frac{\cot (u)}{3} + C$

Step 10

Replace all occurrences of $u$ with $3 x$ .

$4 x^{3} - \frac{\cot (3 x)}{3} + C$

Step 11

Reorder terms.

$4 x^{3} - \frac{1}{3} \cot (3 x) + C$