Calculus Examples

$(4 x^{3} - \csc (2 x + 3) \cot (2 x + 3) - \sqrt[5]{6 - 5 x}) d x$

Step 1

Remove parentheses.

$\int 4 x^{3} - \csc (2 x + 3) \cot (2 x + 3) - \sqrt[5]{6 - 5 x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 4 x^{3} d x + \int - \csc (2 x + 3) \cot (2 x + 3) d x + \int - \sqrt[5]{6 - 5 x} d x$

Step 3

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

$4 \int x^{3} d x + \int - \csc (2 x + 3) \cot (2 x + 3) d x + \int - \sqrt[5]{6 - 5 x} d x$

Step 4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$4 (\frac{1}{4} x^{4} + C) + \int - \csc (2 x + 3) \cot (2 x + 3) d x + \int - \sqrt[5]{6 - 5 x} d x$

Step 5

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

$4 (\frac{1}{4} x^{4} + C) - \int \csc (2 x + 3) \cot (2 x + 3) d x + \int - \sqrt[5]{6 - 5 x} d x$

Step 6

Let $u_{1} = 2 x + 3$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 6.1

Let $u_{1} = 2 x + 3$ . Find $\frac{d u_{1}}{d x}$ .

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Step 6.1.1

Differentiate $2 x + 3$ .

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

Step 6.1.2

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

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

Step 6.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 6.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 6.1.3.2

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

$2 \cdot 1 + \frac{d}{d x} [3]$

Step 6.1.3.3

Multiply $2$ by $1$ .

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

Step 6.1.4

Differentiate using the Constant Rule.

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Step 6.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 + 0$

Step 6.1.4.2

Add $2$ and $0$ .

$2$

Step 6.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$4 (\frac{1}{4} x^{4} + C) - \int \csc (u_{1}) \cot (u_{1}) \frac{1}{2} d u_{1} + \int - \sqrt[5]{6 - 5 x} d x$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$4 (\frac{x^{4}}{4} + C) - \int \csc (u_{1}) \cot (u_{1}) \frac{1}{2} d u_{1} + \int - \sqrt[5]{6 - 5 x} d x$

Step 7.2

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

$4 (\frac{x^{4}}{4} + C) - \int \frac{\csc (u_{1})}{2} \cot (u_{1}) d u_{1} + \int - \sqrt[5]{6 - 5 x} d x$

Step 7.3

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

$4 (\frac{x^{4}}{4} + C) - \int \frac{\csc (u_{1}) \cot (u_{1})}{2} d u_{1} + \int - \sqrt[5]{6 - 5 x} d x$

Step 8

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

$4 (\frac{x^{4}}{4} + C) - (\frac{1}{2} \int \csc (u_{1}) \cot (u_{1}) d u_{1}) + \int - \sqrt[5]{6 - 5 x} d x$

Step 9

Since the derivative of $- \csc (u_{1})$ is $\csc (u_{1}) \cot (u_{1})$ , the integral of $\csc (u_{1}) \cot (u_{1})$ is $- \csc (u_{1})$ .

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) + \int - \sqrt[5]{6 - 5 x} d x$

Step 10

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

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) - \int \sqrt[5]{6 - 5 x} d x$

Step 11

Let $u_{2} = 6 - 5 x$ . Then $d u_{2} = - 5 d x$ , so $- \frac{1}{5} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 11.1

Let $u_{2} = 6 - 5 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 11.1.1

Differentiate $6 - 5 x$ .

$\frac{d}{d x} [6 - 5 x]$

Step 11.1.2

Differentiate.

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Step 11.1.2.1

By the Sum Rule, the derivative of $6 - 5 x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [6] + \frac{d}{d x} [- 5 x]$

Step 11.1.2.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 5 x]$

Step 11.1.3

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 11.1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$0 - 5 \frac{d}{d x} [x]$

Step 11.1.3.2

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

$0 - 5 \cdot 1$

Step 11.1.3.3

Multiply $- 5$ by $1$ .

$0 - 5$

Step 11.1.4

Subtract $5$ from $0$ .

$- 5$

Step 11.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) - \int \sqrt[5]{u_{2}} \frac{1}{- 5} d u_{2}$

Step 12

Simplify.

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Step 12.1

Move the negative in front of the fraction.

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) - \int \sqrt[5]{u_{2}} (- \frac{1}{5}) d u_{2}$

Step 12.2

Combine $\sqrt[5]{u_{2}}$ and $\frac{1}{5}$ .

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) - \int - \frac{\sqrt[5]{u_{2}}}{5} d u_{2}$

Step 13

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) - - \int \frac{\sqrt[5]{u_{2}}}{5} d u_{2}$

Step 14

Simplify.

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Step 14.1

Multiply $- 1$ by $- 1$ .

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) + 1 \int \frac{\sqrt[5]{u_{2}}}{5} d u_{2}$

Step 14.2

Multiply $\int \frac{\sqrt[5]{u_{2}}}{5} d u_{2}$ by $1$ .

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) + \int \frac{\sqrt[5]{u_{2}}}{5} d u_{2}$

Step 15

Since $\frac{1}{5}$ is constant with respect to $u_{2}$ , move $\frac{1}{5}$ out of the integral.

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) + \frac{1}{5} \int \sqrt[5]{u_{2}} d u_{2}$

Step 16

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{u_{2}}$ as ${u_{2}}^{\frac{1}{5}}$ .

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) + \frac{1}{5} \int {u_{2}}^{\frac{1}{5}} d u_{2}$

Step 17

By the Power Rule, the integral of ${u_{2}}^{\frac{1}{5}}$ with respect to $u_{2}$ is $\frac{5}{6} {u_{2}}^{\frac{6}{5}}$ .

$4 (\frac{x^{4}}{4} + C) - \frac{1}{2} (- \csc (u_{1}) + C) + \frac{1}{5} (\frac{5}{6} {u_{2}}^{\frac{6}{5}} + C)$

Step 18

Simplify.

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Step 18.1

Simplify.

$x^{4} + \frac{\csc (u_{1})}{2} + \frac{1}{5} (\frac{5}{6} {u_{2}}^{\frac{6}{5}}) + C$

Step 18.2

Simplify.

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Step 18.2.1

Combine $\frac{5}{6}$ and ${u_{2}}^{\frac{6}{5}}$ .

$x^{4} + \frac{\csc (u_{1})}{2} + \frac{1}{5} \cdot \frac{5 {u_{2}}^{\frac{6}{5}}}{6} + C$

Step 18.2.2

Multiply $\frac{1}{5}$ by $\frac{5 {u_{2}}^{\frac{6}{5}}}{6}$ .

$x^{4} + \frac{\csc (u_{1})}{2} + \frac{5 {u_{2}}^{\frac{6}{5}}}{5 \cdot 6} + C$

Step 18.2.3

Multiply $5$ by $6$ .

$x^{4} + \frac{\csc (u_{1})}{2} + \frac{5 {u_{2}}^{\frac{6}{5}}}{30} + C$

Step 18.2.4

Factor $5$ out of $5 {u_{2}}^{\frac{6}{5}}$ .

$x^{4} + \frac{\csc (u_{1})}{2} + \frac{5 ({u_{2}}^{\frac{6}{5}})}{30} + C$

Step 18.2.5

Cancel the common factors.

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Step 18.2.5.1

Factor $5$ out of $30$ .

$x^{4} + \frac{\csc (u_{1})}{2} + \frac{5 {u_{2}}^{\frac{6}{5}}}{5 \cdot 6} + C$

Step 18.2.5.2

Cancel the common factor.

$x^{4} + \frac{\csc (u_{1})}{2} + \frac{5 {u_{2}}^{\frac{6}{5}}}{5 \cdot 6} + C$

Step 18.2.5.3

Rewrite the expression.

$x^{4} + \frac{\csc (u_{1})}{2} + \frac{{u_{2}}^{\frac{6}{5}}}{6} + C$

Step 19

Substitute back in for each integration substitution variable.

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Step 19.1

Replace all occurrences of $u_{1}$ with $2 x + 3$ .

$x^{4} + \frac{\csc (2 x + 3)}{2} + \frac{{u_{2}}^{\frac{6}{5}}}{6} + C$

Step 19.2

Replace all occurrences of $u_{2}$ with $6 - 5 x$ .

$x^{4} + \frac{\csc (2 x + 3)}{2} + \frac{{(6 - 5 x)}^{\frac{6}{5}}}{6} + C$

Step 20

Reorder terms.

$x^{4} + \frac{1}{2} \csc (2 x + 3) + \frac{1}{6} {(6 - 5 x)}^{\frac{6}{5}} + C$