Calculus Examples

$\frac{d y}{d x} + 5 y = 7 x^{2}$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 5$ .

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Step 1.1

Set up the integration.

$e^{\int 5 d x}$

Step 1.2

Apply the constant rule.

$e^{5 x + C}$

Step 1.3

Remove the constant of integration.

$e^{5 x}$

Step 2

Multiply each term by the integrating factor $e^{5 x}$ .

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Step 2.1

Multiply each term by $e^{5 x}$ .

$e^{5 x} \frac{d y}{d x} + e^{5 x} (5 y) = e^{5 x} (7 x^{2})$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{5 x} \frac{d y}{d x} + 5 e^{5 x} y = e^{5 x} (7 x^{2})$

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{5 x} \frac{d y}{d x} + 5 e^{5 x} y = 7 e^{5 x} x^{2}$

Step 2.4

Reorder factors in $e^{5 x} \frac{d y}{d x} + 5 e^{5 x} y = 7 e^{5 x} x^{2}$ .

$e^{5 x} \frac{d y}{d x} + 5 y e^{5 x} = 7 x^{2} e^{5 x}$

Step 3

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [e^{5 x} y] = 7 x^{2} e^{5 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{5 x} y] d x = \int 7 x^{2} e^{5 x} d x$

Step 5

Integrate the left side.

$e^{5 x} y = \int 7 x^{2} e^{5 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$e^{5 x} y = 7 \int x^{2} e^{5 x} d x$

Step 6.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = e^{5 x}$ .

$e^{5 x} y = 7 (x^{2} (\frac{1}{5} e^{5 x}) - \int \frac{1}{5} e^{5 x} (2 x) d x)$

Step 6.3

Simplify.

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Step 6.3.1

Combine $\frac{1}{5}$ and $e^{5 x}$ .

$e^{5 x} y = 7 (x^{2} \frac{e^{5 x}}{5} - \int \frac{1}{5} e^{5 x} (2 x) d x)$

Step 6.3.2

Combine $x^{2}$ and $\frac{e^{5 x}}{5}$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \int \frac{1}{5} e^{5 x} (2 x) d x)$

Step 6.3.3

Combine $\frac{1}{5}$ and $e^{5 x}$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \int \frac{e^{5 x}}{5} (2 x) d x)$

Step 6.3.4

Combine $2$ and $\frac{e^{5 x}}{5}$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \int \frac{2 e^{5 x}}{5} x d x)$

Step 6.3.5

Combine $\frac{2 e^{5 x}}{5}$ and $x$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \int \frac{2 e^{5 x} x}{5} d x)$

Step 6.4

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

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - (\frac{2}{5} \int e^{5 x} x d x))$

Step 6.5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{5 x}$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (x (\frac{1}{5} e^{5 x}) - \int \frac{1}{5} e^{5 x} d x))$

Step 6.6

Simplify.

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Step 6.6.1

Combine $\frac{1}{5}$ and $e^{5 x}$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (x \frac{e^{5 x}}{5} - \int \frac{1}{5} e^{5 x} d x))$

Step 6.6.2

Combine $x$ and $\frac{e^{5 x}}{5}$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - \int \frac{1}{5} e^{5 x} d x))$

Step 6.6.3

Combine $\frac{1}{5}$ and $e^{5 x}$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - \int \frac{e^{5 x}}{5} d x))$

Step 6.7

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

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - (\frac{1}{5} \int e^{5 x} d x)))$

Step 6.8

Let $u = 5 x$ . Then $d u = 5 d x$ , so $\frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.8.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

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Step 6.8.1.1

Differentiate $5 x$ .

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

Step 6.8.1.2

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

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

Step 6.8.1.3

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

$5 \cdot 1$

Step 6.8.1.4

Multiply $5$ by $1$ .

$5$

Step 6.8.2

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

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - \frac{1}{5} \int e^{u} \frac{1}{5} d u))$

Step 6.9

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

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - \frac{1}{5} \int \frac{e^{u}}{5} d u))$

Step 6.10

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

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - \frac{1}{5} (\frac{1}{5} \int e^{u} d u)))$

Step 6.11

Simplify.

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Step 6.11.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - \frac{1}{5 \cdot 5} \int e^{u} d u))$

Step 6.11.2

Multiply $5$ by $5$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - \frac{1}{25} \int e^{u} d u))$

Step 6.12

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - \frac{1}{25} (e^{u} + C)))$

Step 6.13

Rewrite $7 (\frac{x^{2} e^{5 x}}{5} - \frac{2}{5} (\frac{x e^{5 x}}{5} - \frac{1}{25} (e^{u} + C)))$ as $7 (\frac{x^{2} e^{5 x}}{5} - \frac{2 x e^{5 x}}{25} + \frac{2 e^{u}}{125}) + C$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2 x e^{5 x}}{25} + \frac{2 e^{u}}{125}) + C$

Step 6.14

Replace all occurrences of $u$ with $5 x$ .

$e^{5 x} y = 7 (\frac{x^{2} e^{5 x}}{5} - \frac{2 x e^{5 x}}{25} + \frac{2 e^{5 x}}{125}) + C$

Step 6.15

Reorder terms.

$e^{5 x} y = 7 (\frac{1}{5} x^{2} e^{5 x} - \frac{2}{25} x e^{5 x} + \frac{2}{125} e^{5 x}) + C$

Step 7

Solve for $y$ .

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Step 7.1

Simplify.

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Step 7.1.1

Combine $x$ and $\frac{2}{25}$ .

$e^{5 x} y = 7 (\frac{1}{5} \cdot (x^{2} e^{5 x}) - \frac{x \cdot 2}{25} \cdot e^{5 x} + \frac{2}{125} \cdot e^{5 x}) + C$

Step 7.1.2

Combine $e^{5 x}$ and $\frac{x \cdot 2}{25}$ .

$e^{5 x} y = 7 (\frac{1}{5} \cdot (x^{2} e^{5 x}) - \frac{e^{5 x} (x \cdot 2)}{25} + \frac{2}{125} \cdot e^{5 x}) + C$

Step 7.1.3

Remove parentheses.

$e^{5 x} y = 7 (\frac{1}{5} \cdot x^{2} e^{5 x} - \frac{e^{5 x} (x \cdot 2)}{25} + \frac{2}{125} \cdot e^{5 x}) + C$

Step 7.2

Divide each term in $e^{5 x} y = 7 (\frac{1}{5} \cdot (x^{2} e^{5 x}) - \frac{e^{5 x} (x \cdot 2)}{25} + \frac{2}{125} \cdot e^{5 x}) + C$ by $e^{5 x}$ and simplify.

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Step 7.2.1

Divide each term in $e^{5 x} y = 7 (\frac{1}{5} \cdot (x^{2} e^{5 x}) - \frac{e^{5 x} (x \cdot 2)}{25} + \frac{2}{125} \cdot e^{5 x}) + C$ by $e^{5 x}$ .

$\frac{e^{5 x} y}{e^{5 x}} = \frac{7 (\frac{1}{5} \cdot (x^{2} e^{5 x}) - \frac{e^{5 x} (x \cdot 2)}{25} + \frac{2}{125} \cdot e^{5 x})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $e^{5 x}$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{e^{5 x} y}{e^{5 x}} = \frac{7 (\frac{1}{5} \cdot (x^{2} e^{5 x}) - \frac{e^{5 x} (x \cdot 2)}{25} + \frac{2}{125} \cdot e^{5 x})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{7 (\frac{1}{5} \cdot (x^{2} e^{5 x}) - \frac{e^{5 x} (x \cdot 2)}{25} + \frac{2}{125} \cdot e^{5 x})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Simplify each term.

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Step 7.2.3.1.1

Simplify the numerator.

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Step 7.2.3.1.1.1

Factor $e^{5 x}$ out of $\frac{1}{5} \cdot x^{2} e^{5 x} - \frac{e^{5 x} x \cdot 2}{25} + \frac{2}{125} \cdot e^{5 x}$ .

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Step 7.2.3.1.1.1.1

Factor $e^{5 x}$ out of $\frac{1}{5} \cdot x^{2} e^{5 x}$ .

$y = \frac{7 (e^{5 x} (\frac{1}{5} \cdot x^{2}) - \frac{e^{5 x} x \cdot 2}{25} + \frac{2}{125} \cdot e^{5 x})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.1.2

Factor $e^{5 x}$ out of $- \frac{e^{5 x} x \cdot 2}{25}$ .

$y = \frac{7 (e^{5 x} (\frac{1}{5} \cdot x^{2}) + e^{5 x} (- \frac{x \cdot 2}{25}) + \frac{2}{125} \cdot e^{5 x})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.1.3

Factor $e^{5 x}$ out of $\frac{2}{125} \cdot e^{5 x}$ .

$y = \frac{7 (e^{5 x} (\frac{1}{5} \cdot x^{2}) + e^{5 x} (- \frac{x \cdot 2}{25}) + e^{5 x} \cdot \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.1.4

Factor $e^{5 x}$ out of $e^{5 x} (\frac{1}{5} \cdot x^{2}) + e^{5 x} (- \frac{x \cdot 2}{25})$ .

$y = \frac{7 (e^{5 x} (\frac{1}{5} \cdot x^{2} - \frac{x \cdot 2}{25}) + e^{5 x} \cdot \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.1.5

Factor $e^{5 x}$ out of $e^{5 x} (\frac{1}{5} \cdot x^{2} - \frac{x \cdot 2}{25}) + e^{5 x} \cdot \frac{2}{125}$ .

$y = \frac{7 (e^{5 x} (\frac{1}{5} \cdot x^{2} - \frac{x \cdot 2}{25} + \frac{2}{125}))}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.2

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

$y = \frac{7 (e^{5 x} (\frac{x^{2}}{5} - \frac{x \cdot 2}{25} + \frac{2}{125}))}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.3

Move $2$ to the left of $x$ .

$y = \frac{7 (e^{5 x} (\frac{x^{2}}{5} - \frac{2 \cdot x}{25} + \frac{2}{125}))}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.4

Remove unnecessary parentheses.

$y = \frac{7 e^{5 x} (\frac{x^{2}}{5} - \frac{2 x}{25} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.5

To write $\frac{x^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{7 e^{5 x} (\frac{x^{2}}{5} \cdot \frac{5}{5} - \frac{2 x}{25} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.6

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.1.1.6.1

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

$y = \frac{7 e^{5 x} (\frac{x^{2} \cdot 5}{5 \cdot 5} - \frac{2 x}{25} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.6.2

Multiply $5$ by $5$ .

$y = \frac{7 e^{5 x} (\frac{x^{2} \cdot 5}{25} - \frac{2 x}{25} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.7

Combine the numerators over the common denominator.

$y = \frac{7 e^{5 x} (\frac{x^{2} \cdot 5 - 2 x}{25} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.8

Simplify the numerator.

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Step 7.2.3.1.1.8.1

Factor $x$ out of $x^{2} \cdot 5 - 2 x$ .

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Step 7.2.3.1.1.8.1.1

Factor $x$ out of $x^{2} \cdot 5$ .

$y = \frac{7 e^{5 x} (\frac{x (x \cdot 5) - 2 x}{25} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.8.1.2

Factor $x$ out of $- 2 x$ .

$y = \frac{7 e^{5 x} (\frac{x (x \cdot 5) + x \cdot - 2}{25} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.8.1.3

Factor $x$ out of $x (x \cdot 5) + x \cdot - 2$ .

$y = \frac{7 e^{5 x} (\frac{x (x \cdot 5 - 2)}{25} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.8.2

Move $5$ to the left of $x$ .

$y = \frac{7 e^{5 x} (\frac{x (5 x - 2)}{25} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.9

To write $\frac{x (5 x - 2)}{25}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{7 e^{5 x} (\frac{x (5 x - 2)}{25} \cdot \frac{5}{5} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.10

Write each expression with a common denominator of $125$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.1.1.10.1

Multiply $\frac{x (5 x - 2)}{25}$ by $\frac{5}{5}$ .

$y = \frac{7 e^{5 x} (\frac{x (5 x - 2) \cdot 5}{25 \cdot 5} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.10.2

Multiply $25$ by $5$ .

$y = \frac{7 e^{5 x} (\frac{x (5 x - 2) \cdot 5}{125} + \frac{2}{125})}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.11

Combine the numerators over the common denominator.

$y = \frac{7 e^{5 x} \frac{x (5 x - 2) \cdot 5 + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.12

Simplify the numerator.

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Step 7.2.3.1.1.12.1

Apply the distributive property.

$y = \frac{7 e^{5 x} \frac{(x (5 x) + x \cdot - 2) \cdot 5 + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.12.2

Rewrite using the commutative property of multiplication.

$y = \frac{7 e^{5 x} \frac{(5 x \cdot x + x \cdot - 2) \cdot 5 + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.12.3

Move $- 2$ to the left of $x$ .

$y = \frac{7 e^{5 x} \frac{(5 x \cdot x - 2 \cdot x) \cdot 5 + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.12.4

Multiply $x$ by $x$ by adding the exponents.

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Step 7.2.3.1.1.12.4.1

Move $x$ .

$y = \frac{7 e^{5 x} \frac{(5 (x \cdot x) - 2 \cdot x) \cdot 5 + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.12.4.2

Multiply $x$ by $x$ .

$y = \frac{7 e^{5 x} \frac{(5 x^{2} - 2 \cdot x) \cdot 5 + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

$y = \frac{7 e^{5 x} \frac{(5 x^{2} - 2 x) \cdot 5 + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.12.5

Apply the distributive property.

$y = \frac{7 e^{5 x} \frac{5 x^{2} \cdot 5 - 2 x \cdot 5 + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.12.6

Multiply $5$ by $5$ .

$y = \frac{7 e^{5 x} \frac{25 x^{2} - 2 x \cdot 5 + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.12.7

Multiply $5$ by $- 2$ .

$y = \frac{7 e^{5 x} \frac{25 x^{2} - 10 x + 2}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.13

Combine exponents.

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Step 7.2.3.1.1.13.1

Combine $\frac{25 x^{2} - 10 x + 2}{125}$ and $7$ .

$y = \frac{\frac{(25 x^{2} - 10 x + 2) \cdot 7}{125} e^{5 x}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.13.2

Combine $\frac{(25 x^{2} - 10 x + 2) \cdot 7}{125}$ and $e^{5 x}$ .

$y = \frac{\frac{(25 x^{2} - 10 x + 2) \cdot 7 e^{5 x}}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.1.14

Move $7$ to the left of $25 x^{2} - 10 x + 2$ .

$y = \frac{\frac{7 (25 x^{2} - 10 x + 2) e^{5 x}}{125}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{7 (25 x^{2} - 10 x + 2) e^{5 x}}{125} \cdot \frac{1}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.3

Combine.

$y = \frac{7 (25 x^{2} - 10 x + 2) e^{5 x} \cdot 1}{125 e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.4

Cancel the common factor of $e^{5 x}$ .

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Step 7.2.3.1.4.1

Cancel the common factor.

$y = \frac{7 (25 x^{2} - 10 x + 2) e^{5 x} \cdot 1}{125 e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.4.2

Rewrite the expression.

$y = \frac{7 (25 x^{2} - 10 x + 2) \cdot 1}{125} + \frac{C}{e^{5 x}}$

Step 7.2.3.1.5

Multiply $7$ by $1$ .

$y = \frac{7 (25 x^{2} - 10 x + 2)}{125} + \frac{C}{e^{5 x}}$

Step 7.2.3.2

To write $\frac{7 (25 x^{2} - 10 x + 2)}{125}$ as a fraction with a common denominator, multiply by $\frac{e^{5 x}}{e^{5 x}}$ .

$y = \frac{7 (25 x^{2} - 10 x + 2)}{125} \cdot \frac{e^{5 x}}{e^{5 x}} + \frac{C}{e^{5 x}}$

Step 7.2.3.3

To write $\frac{C}{e^{5 x}}$ as a fraction with a common denominator, multiply by $\frac{125}{125}$ .

$y = \frac{7 (25 x^{2} - 10 x + 2)}{125} \cdot \frac{e^{5 x}}{e^{5 x}} + \frac{C}{e^{5 x}} \cdot \frac{125}{125}$

Step 7.2.3.4

Write each expression with a common denominator of $125 e^{5 x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.4.1

Multiply $\frac{7 (25 x^{2} - 10 x + 2)}{125}$ by $\frac{e^{5 x}}{e^{5 x}}$ .

$y = \frac{7 (25 x^{2} - 10 x + 2) e^{5 x}}{125 e^{5 x}} + \frac{C}{e^{5 x}} \cdot \frac{125}{125}$

Step 7.2.3.4.2

Multiply $\frac{C}{e^{5 x}}$ by $\frac{125}{125}$ .

$y = \frac{7 (25 x^{2} - 10 x + 2) e^{5 x}}{125 e^{5 x}} + \frac{C \cdot 125}{e^{5 x} \cdot 125}$

Step 7.2.3.4.3

Reorder the factors of $e^{5 x} \cdot 125$ .

$y = \frac{7 (25 x^{2} - 10 x + 2) e^{5 x}}{125 e^{5 x}} + \frac{C \cdot 125}{125 e^{5 x}}$

Step 7.2.3.5

Combine the numerators over the common denominator.

$y = \frac{7 (25 x^{2} - 10 x + 2) e^{5 x} + C \cdot 125}{125 e^{5 x}}$

Step 7.2.3.6

Simplify the numerator.

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Step 7.2.3.6.1

Apply the distributive property.

$y = \frac{(7 (25 x^{2}) + 7 (- 10 x) + 7 \cdot 2) e^{5 x} + C \cdot 125}{125 e^{5 x}}$

Step 7.2.3.6.2

Simplify.

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Step 7.2.3.6.2.1

Multiply $25$ by $7$ .

$y = \frac{(175 x^{2} + 7 (- 10 x) + 7 \cdot 2) e^{5 x} + C \cdot 125}{125 e^{5 x}}$

Step 7.2.3.6.2.2

Multiply $- 10$ by $7$ .

$y = \frac{(175 x^{2} - 70 x + 7 \cdot 2) e^{5 x} + C \cdot 125}{125 e^{5 x}}$

Step 7.2.3.6.2.3

Multiply $7$ by $2$ .

$y = \frac{(175 x^{2} - 70 x + 14) e^{5 x} + C \cdot 125}{125 e^{5 x}}$

Step 7.2.3.6.3

Apply the distributive property.

$y = \frac{175 x^{2} e^{5 x} - 70 x e^{5 x} + 14 e^{5 x} + C \cdot 125}{125 e^{5 x}}$

Step 7.2.3.6.4

Move $125$ to the left of $C$ .

$y = \frac{175 x^{2} e^{5 x} - 70 x e^{5 x} + 14 e^{5 x} + 125 C}{125 e^{5 x}}$