Calculus Examples

$\frac{d y}{d x} = x {(2 x - 1)}^{5}$

Step 1

Rewrite the equation.

$d y = x {(2 x - 1)}^{5} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x {(2 x - 1)}^{5} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int x {(2 x - 1)}^{5} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Let $u = 2 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $2 x - 1$ .

$\frac{d}{d x} [2 x - 1]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

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

Step 2.3.1.1.3

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

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Step 2.3.1.1.3.1

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

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

Step 2.3.1.1.3.2

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

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

Step 2.3.1.1.3.3

Multiply $2$ by $1$ .

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

Step 2.3.1.1.4

Differentiate using the Constant Rule.

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Step 2.3.1.1.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 + 0$

Step 2.3.1.1.4.2

Add $2$ and $0$ .

$2$

Step 2.3.1.2

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

$y + C_{1} = \int (\frac{u}{2} + \frac{1}{2}) u^{5} \frac{1}{2} d u$

Step 2.3.2

Combine $\frac{1}{2}$ and $u^{5}$ .

$y + C_{1} = \int (\frac{u}{2} + \frac{1}{2}) \frac{u^{5}}{2} d u$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Apply the distributive property.

$y + C_{1} = \int \frac{u}{2} \cdot \frac{u^{5}}{2} + \frac{1}{2} \cdot \frac{u^{5}}{2} d u$

Step 2.3.3.2

Multiply $\frac{u}{2}$ by $\frac{u^{5}}{2}$ .

$y + C_{1} = \int \frac{u \cdot u^{5}}{2 \cdot 2} + \frac{1}{2} \cdot \frac{u^{5}}{2} d u$

Step 2.3.3.3

Raise $u$ to the power of $1$ .

$y + C_{1} = \int \frac{u^{1} u^{5}}{2 \cdot 2} + \frac{1}{2} \cdot \frac{u^{5}}{2} d u$

Step 2.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y + C_{1} = \int \frac{u^{1 + 5}}{2 \cdot 2} + \frac{1}{2} \cdot \frac{u^{5}}{2} d u$

Step 2.3.3.5

Add $1$ and $5$ .

$y + C_{1} = \int \frac{u^{6}}{2 \cdot 2} + \frac{1}{2} \cdot \frac{u^{5}}{2} d u$

Step 2.3.3.6

Multiply $2$ by $2$ .

$y + C_{1} = \int \frac{u^{6}}{4} + \frac{1}{2} \cdot \frac{u^{5}}{2} d u$

Step 2.3.3.7

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

$y + C_{1} = \int \frac{u^{6}}{4} + \frac{u^{5}}{2 \cdot 2} d u$

Step 2.3.3.8

Multiply $2$ by $2$ .

$y + C_{1} = \int \frac{u^{6}}{4} + \frac{u^{5}}{4} d u$

Step 2.3.4

Split the single integral into multiple integrals.

$y + C_{1} = \int \frac{u^{6}}{4} d u + \int \frac{u^{5}}{4} d u$

Step 2.3.5

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$y + C_{1} = \frac{1}{4} \int u^{6} d u + \int \frac{u^{5}}{4} d u$

Step 2.3.6

By the Power Rule, the integral of $u^{6}$ with respect to $u$ is $\frac{1}{7} u^{7}$ .

$y + C_{1} = \frac{1}{4} (\frac{1}{7} u^{7} + C_{2}) + \int \frac{u^{5}}{4} d u$

Step 2.3.7

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$y + C_{1} = \frac{1}{4} (\frac{1}{7} u^{7} + C_{2}) + \frac{1}{4} \int u^{5} d u$

Step 2.3.8

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

$y + C_{1} = \frac{1}{4} (\frac{1}{7} u^{7} + C_{2}) + \frac{1}{4} (\frac{1}{6} u^{6} + C_{3})$

Step 2.3.9

Simplify.

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Step 2.3.9.1

Simplify.

$y + C_{1} = \frac{1}{4} \cdot \frac{1}{7} u^{7} + \frac{1}{4} \cdot \frac{1}{6} u^{6} + C_{4}$

Step 2.3.9.2

Simplify.

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Step 2.3.9.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{7}$ .

$y + C_{1} = \frac{1}{4 \cdot 7} u^{7} + \frac{1}{4} \cdot \frac{1}{6} u^{6} + C_{4}$

Step 2.3.9.2.2

Multiply $4$ by $7$ .

$y + C_{1} = \frac{1}{28} u^{7} + \frac{1}{4} \cdot \frac{1}{6} u^{6} + C_{4}$

Step 2.3.9.2.3

Multiply $\frac{1}{4}$ by $\frac{1}{6}$ .

$y + C_{1} = \frac{1}{28} u^{7} + \frac{1}{4 \cdot 6} u^{6} + C_{4}$

Step 2.3.9.2.4

Multiply $4$ by $6$ .

$y + C_{1} = \frac{1}{28} u^{7} + \frac{1}{24} u^{6} + C_{4}$

Step 2.3.10

Replace all occurrences of $u$ with $2 x - 1$ .

$y + C_{1} = \frac{1}{28} {(2 x - 1)}^{7} + \frac{1}{24} {(2 x - 1)}^{6} + C_{4}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = \frac{1}{28} {(2 x - 1)}^{7} + \frac{1}{24} {(2 x - 1)}^{6} + K$