Calculus Examples

$t \frac{d y}{d x} = 6 t e^{2 t} + x (2 t - 1)$

Step 1

Rewrite the equation.

$t d y = (6 t e^{2 t} + x (2 t - 1)) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int t d y = \int 6 t e^{2 t} + x (2 t - 1) d x$

Step 2.2

Apply the constant rule.

$t y + C_{1} = \int 6 t e^{2 t} + x (2 t - 1) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$t y + C_{1} = \int 6 t e^{2 t} d x + \int x (2 t - 1) d x$

Step 2.3.2

Apply the constant rule.

$t y + C_{1} = 6 t e^{2 t} x + C_{2} + \int x (2 t - 1) d x$

Step 2.3.3

Since $2 t - 1$ is constant with respect to $x$ , move $2 t - 1$ out of the integral.

$t y + C_{1} = 6 t e^{2 t} x + C_{2} + (2 t - 1) \int x d x$

Step 2.3.4

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

$t y + C_{1} = 6 t e^{2 t} x + C_{2} + (2 t - 1) (\frac{1}{2} x^{2} + C_{3})$

Step 2.3.5

Simplify.

$t y + C_{1} = 6 t e^{2 t} x + (2 t - 1) \frac{1}{2} x^{2} + C_{4}$

Step 2.3.6

Reorder terms.

$t y + C_{1} = 6 e^{2 t} t x + \frac{1}{2} x^{2} (2 t - 1) + C_{4}$

Step 2.4

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

$t y = 6 e^{2 t} t x + \frac{1}{2} x^{2} (2 t - 1) + K$

Step 3

Divide each term in $t y = 6 e^{2 t} t x + \frac{1}{2} \cdot (x^{2} (2 t - 1)) + K$ by $t$ and simplify.

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Step 3.1

Divide each term in $t y = 6 e^{2 t} t x + \frac{1}{2} \cdot (x^{2} (2 t - 1)) + K$ by $t$ .

$\frac{t y}{t} = \frac{6 e^{2 t} t x}{t} + \frac{\frac{1}{2} \cdot (x^{2} (2 t - 1))}{t} + \frac{K}{t}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $t$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{t y}{t} = \frac{6 e^{2 t} t x}{t} + \frac{\frac{1}{2} \cdot (x^{2} (2 t - 1))}{t} + \frac{K}{t}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{6 e^{2 t} t x}{t} + \frac{\frac{1}{2} \cdot (x^{2} (2 t - 1))}{t} + \frac{K}{t}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Combine the numerators over the common denominator.

$y = \frac{6 e^{2 t} t x + \frac{1}{2} \cdot (x^{2} (2 t - 1)) + K}{t}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Apply the distributive property.

$y = \frac{6 e^{2 t} t x + \frac{1}{2} \cdot (x^{2} (2 t) + x^{2} \cdot - 1) + K}{t}$

Step 3.3.2.2

Rewrite using the commutative property of multiplication.

$y = \frac{6 e^{2 t} t x + \frac{1}{2} \cdot (2 x^{2} t + x^{2} \cdot - 1) + K}{t}$

Step 3.3.2.3

Move $- 1$ to the left of $x^{2}$ .

$y = \frac{6 e^{2 t} t x + \frac{1}{2} \cdot (2 x^{2} t - 1 \cdot x^{2}) + K}{t}$

Step 3.3.2.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$y = \frac{6 e^{2 t} t x + \frac{1}{2} \cdot (2 x^{2} t - x^{2}) + K}{t}$

Step 3.3.2.5

Apply the distributive property.

$y = \frac{6 e^{2 t} t x + \frac{1}{2} (2 x^{2} t) + \frac{1}{2} (- x^{2}) + K}{t}$

Step 3.3.2.6

Cancel the common factor of $2$ .

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Step 3.3.2.6.1

Factor $2$ out of $2 x^{2} t$ .

$y = \frac{6 e^{2 t} t x + \frac{1}{2} (2 (x^{2} t)) + \frac{1}{2} (- x^{2}) + K}{t}$

Step 3.3.2.6.2

Cancel the common factor.

$y = \frac{6 e^{2 t} t x + \frac{1}{2} (2 (x^{2} t)) + \frac{1}{2} (- x^{2}) + K}{t}$

Step 3.3.2.6.3

Rewrite the expression.

$y = \frac{6 e^{2 t} t x + x^{2} t + \frac{1}{2} (- x^{2}) + K}{t}$

Step 3.3.2.7

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

$y = \frac{6 e^{2 t} t x + x^{2} t - \frac{x^{2}}{2} + K}{t}$

Step 3.3.2.8

Combine $x^{2} t$ and $- \frac{x^{2}}{2}$ using a common denominator.

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Step 3.3.2.8.1

Reorder $x^{2}$ and $t$ .

$y = \frac{6 e^{2 t} t x + t x^{2} - \frac{x^{2}}{2} + K}{t}$

Step 3.3.2.8.2

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

$y = \frac{6 e^{2 t} t x + t x^{2} \cdot \frac{2}{2} - \frac{x^{2}}{2} + K}{t}$

Step 3.3.2.8.3

Combine $t x^{2}$ and $\frac{2}{2}$ .

$y = \frac{6 e^{2 t} t x + \frac{t x^{2} \cdot 2}{2} - \frac{x^{2}}{2} + K}{t}$

Step 3.3.2.8.4

Combine the numerators over the common denominator.

$y = \frac{6 e^{2 t} t x + \frac{t x^{2} \cdot 2 - x^{2}}{2} + K}{t}$

Step 3.3.2.9

Simplify the numerator.

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Step 3.3.2.9.1

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

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Step 3.3.2.9.1.1

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

$y = \frac{6 e^{2 t} t x + \frac{x^{2} (t \cdot 2) - x^{2}}{2} + K}{t}$

Step 3.3.2.9.1.2

Factor $x^{2}$ out of $- x^{2}$ .

$y = \frac{6 e^{2 t} t x + \frac{x^{2} (t \cdot 2) + x^{2} \cdot - 1}{2} + K}{t}$

Step 3.3.2.9.1.3

Factor $x^{2}$ out of $x^{2} (t \cdot 2) + x^{2} \cdot - 1$ .

$y = \frac{6 e^{2 t} t x + \frac{x^{2} (t \cdot 2 - 1)}{2} + K}{t}$

Step 3.3.2.9.2

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

$y = \frac{6 e^{2 t} t x + \frac{x^{2} (2 t - 1)}{2} + K}{t}$

Step 3.3.3

To write $6 e^{2 t} t x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{6 e^{2 t} t x \cdot \frac{2}{2} + \frac{x^{2} (2 t - 1)}{2} + K}{t}$

Step 3.3.4

Simplify terms.

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Step 3.3.4.1

Combine $6 e^{2 t} t x$ and $\frac{2}{2}$ .

$y = \frac{\frac{6 e^{2 t} t x \cdot 2}{2} + \frac{x^{2} (2 t - 1)}{2} + K}{t}$

Step 3.3.4.2

Combine the numerators over the common denominator.

$y = \frac{\frac{6 e^{2 t} t x \cdot 2 + x^{2} (2 t - 1)}{2} + K}{t}$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Factor $x$ out of $6 e^{2 t} t x \cdot 2 + x^{2} (2 t - 1)$ .

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Step 3.3.5.1.1

Factor $x$ out of $6 e^{2 t} t x \cdot 2$ .

$y = \frac{\frac{x (6 e^{2 t} t \cdot 2) + x^{2} (2 t - 1)}{2} + K}{t}$

Step 3.3.5.1.2

Factor $x$ out of $x^{2} (2 t - 1)$ .

$y = \frac{\frac{x (6 e^{2 t} t \cdot 2) + x (x (2 t - 1))}{2} + K}{t}$

Step 3.3.5.1.3

Factor $x$ out of $x (6 e^{2 t} t \cdot 2) + x (x (2 t - 1))$ .

$y = \frac{\frac{x (6 e^{2 t} t \cdot 2 + x (2 t - 1))}{2} + K}{t}$

Step 3.3.5.2

Multiply $2$ by $6$ .

$y = \frac{\frac{x (12 e^{2 t} t + x (2 t - 1))}{2} + K}{t}$

Step 3.3.5.3

Apply the distributive property.

$y = \frac{\frac{x (12 e^{2 t} t + x (2 t) + x \cdot - 1)}{2} + K}{t}$

Step 3.3.5.4

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{x (12 e^{2 t} t + 2 x t + x \cdot - 1)}{2} + K}{t}$

Step 3.3.5.5

Move $- 1$ to the left of $x$ .

$y = \frac{\frac{x (12 e^{2 t} t + 2 x t - 1 \cdot x)}{2} + K}{t}$

Step 3.3.5.6

Simplify each term.

$y = \frac{\frac{x (12 e^{2 t} t + 2 x t - x)}{2} + K}{t}$

Step 3.3.6

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

$y = \frac{\frac{x (12 e^{2 t} t + 2 x t - x)}{2} + K \cdot \frac{2}{2}}{t}$

Step 3.3.7

Simplify terms.

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Step 3.3.7.1

Combine $K$ and $\frac{2}{2}$ .

$y = \frac{\frac{x (12 e^{2 t} t + 2 x t - x)}{2} + \frac{K \cdot 2}{2}}{t}$

Step 3.3.7.2

Combine the numerators over the common denominator.

$y = \frac{\frac{x (12 e^{2 t} t + 2 x t - x) + K \cdot 2}{2}}{t}$

Step 3.3.8

Simplify the numerator.

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Step 3.3.8.1

Apply the distributive property.

$y = \frac{\frac{x (12 e^{2 t} t) + x (2 x t) + x (- x) + K \cdot 2}{2}}{t}$

Step 3.3.8.2

Simplify.

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Step 3.3.8.2.1

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{12 x e^{2 t} t + x (2 x t) + x (- x) + K \cdot 2}{2}}{t}$

Step 3.3.8.2.2

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{12 x e^{2 t} t + 2 x (x t) + x (- x) + K \cdot 2}{2}}{t}$

Step 3.3.8.2.3

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{12 x e^{2 t} t + 2 x (x t) - x \cdot x + K \cdot 2}{2}}{t}$

Step 3.3.8.3

Simplify each term.

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Step 3.3.8.3.1

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

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Step 3.3.8.3.1.1

Move $x$ .

$y = \frac{\frac{12 x e^{2 t} t + 2 (x \cdot x) t - x \cdot x + K \cdot 2}{2}}{t}$

Step 3.3.8.3.1.2

Multiply $x$ by $x$ .

$y = \frac{\frac{12 x e^{2 t} t + 2 x^{2} t - x \cdot x + K \cdot 2}{2}}{t}$

Step 3.3.8.3.2

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

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Step 3.3.8.3.2.1

Move $x$ .

$y = \frac{\frac{12 x e^{2 t} t + 2 x^{2} t - (x \cdot x) + K \cdot 2}{2}}{t}$

Step 3.3.8.3.2.2

Multiply $x$ by $x$ .

$y = \frac{\frac{12 x e^{2 t} t + 2 x^{2} t - x^{2} + K \cdot 2}{2}}{t}$

Step 3.3.8.4

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

$y = \frac{\frac{12 x e^{2 t} t + 2 x^{2} t - x^{2} + 2 K}{2}}{t}$

Step 3.3.9

Reorder factors in $\frac{12 x e^{2 t} t + 2 x^{2} t - x^{2} + 2 K}{2}$ .

$y = \frac{\frac{12 x t e^{2 t} + 2 x^{2} t - x^{2} + 2 K}{2}}{t}$

Step 3.3.10

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{12 x t e^{2 t} + 2 x^{2} t - x^{2} + 2 K}{2} \cdot \frac{1}{t}$

Step 3.3.11

Multiply $\frac{12 x t e^{2 t} + 2 x^{2} t - x^{2} + 2 K}{2}$ by $\frac{1}{t}$ .

$y = \frac{12 x t e^{2 t} + 2 x^{2} t - x^{2} + 2 K}{2 t}$

Step 4

Simplify the constant of integration.

$y = \frac{12 x t e^{2 t} + 2 x^{2} t - x^{2} + K}{2 t}$