Calculus Examples

$\int (6 x^{2} + x) \csc^{2} (4 x^{3} + x^{2}) d x$

Step 1

Let $u = 4 x^{3} + x^{2}$ . Then $d u = (12 x^{2} + 2 x) d x$ , so $\frac{1}{2} d u = (6 x^{2} + x) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 x^{3} + x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x^{3} + x^{2}$ .

$\frac{d}{d x} [4 x^{3} + x^{2}]$

Step 1.1.2

By the Sum Rule, the derivative of $4 x^{3} + x^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 (3 x^{2}) + \frac{d}{d x} [x^{2}]$

Step 1.1.3.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [x^{2}]$

Step 1.1.4

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

$12 x^{2} + 2 x$

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Simplify.

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

Step 5.2

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

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

Step 6

Replace all occurrences of $u$ with $4 x^{3} + x^{2}$ .

$- \frac{\cot (4 x^{3} + x^{2})}{2} + C$

Step 7

Reorder terms.

$- \frac{1}{2} \cot (4 x^{3} + x^{2}) + C$