Calculus Examples

$x e^{- t} \frac{d x}{d t} = t$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Divide each term in $x e^{- t} \frac{d x}{d t} = t$ by $x e^{- t}$ and simplify.

Tap for more steps...

Step 1.1.1

Divide each term in $x e^{- t} \frac{d x}{d t} = t$ by $x e^{- t}$ .

$\frac{x e^{- t} \frac{d x}{d t}}{x e^{- t}} = \frac{t}{x e^{- t}}$

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.1.2.1.1

Cancel the common factor.

$\frac{x e^{- t} \frac{d x}{d t}}{x e^{- t}} = \frac{t}{x e^{- t}}$

Step 1.1.2.1.2

Rewrite the expression.

$\frac{e^{- t} \frac{d x}{d t}}{e^{- t}} = \frac{t}{x e^{- t}}$

Step 1.1.2.2

Cancel the common factor of $e^{- t}$ .

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor.

$\frac{e^{- t} \frac{d x}{d t}}{e^{- t}} = \frac{t}{x e^{- t}}$

Step 1.1.2.2.2

Divide $\frac{d x}{d t}$ by $1$ .

$\frac{d x}{d t} = \frac{t}{x e^{- t}}$

Step 1.2

Regroup factors.

$\frac{d x}{d t} = \frac{t}{e^{- t}} \cdot \frac{1}{x}$

Step 1.3

Multiply both sides by $x$ .

$x \frac{d x}{d t} = x (\frac{t}{e^{- t}} \cdot \frac{1}{x})$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Combine.

$x \frac{d x}{d t} = x \frac{t \cdot 1}{e^{- t} x}$

Step 1.4.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.4.2.1

Factor $x$ out of $e^{- t} x$ .

$x \frac{d x}{d t} = x \frac{t \cdot 1}{x e^{- t}}$

Step 1.4.2.2

Cancel the common factor.

$x \frac{d x}{d t} = x \frac{t \cdot 1}{x e^{- t}}$

Step 1.4.2.3

Rewrite the expression.

$x \frac{d x}{d t} = \frac{t \cdot 1}{e^{- t}}$

Step 1.4.3

Multiply $t$ by $1$ .

$x \frac{d x}{d t} = \frac{t}{e^{- t}}$

Step 1.5

Rewrite the equation.

$x d x = \frac{t}{e^{- t}} d t$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int x d x = \int \frac{t}{e^{- t}} d t$

Step 2.2

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

$\frac{1}{2} x^{2} + C_{1} = \int \frac{t}{e^{- t}} d t$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Simplify the expression.

Tap for more steps...

Step 2.3.1.1

Negate the exponent of $e^{- t}$ and move it out of the denominator.

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

Step 2.3.1.2

Multiply the exponents in ${(e^{- t})}^{- 1}$ .

Tap for more steps...

Step 2.3.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} x^{2} + C_{1} = \int t e^{- t \cdot - 1} d t$

Step 2.3.1.2.2

Multiply $- t \cdot - 1$ .

Tap for more steps...

Step 2.3.1.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} x^{2} + C_{1} = \int t e^{1 t} d t$

Step 2.3.1.2.2.2

Multiply $t$ by $1$ .

$\frac{1}{2} x^{2} + C_{1} = \int t e^{t} d t$

Step 2.3.2

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

$\frac{1}{2} x^{2} + C_{1} = t e^{t} - \int e^{t} d t$

Step 2.3.3

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

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

Step 2.3.4

Simplify.

$\frac{1}{2} x^{2} + C_{1} = t e^{t} - e^{t} + C_{2}$

Step 2.3.5

Reorder terms.

$\frac{1}{2} x^{2} + C_{1} = e^{t} t - e^{t} + C_{2}$

Step 2.4

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

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

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.1.1

Simplify $2 (\frac{1}{2} x^{2})$ .

Tap for more steps...

Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

$x^{2} = 2 (e^{t} t - e^{t} + K)$

Step 3.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.1

Simplify $2 (e^{t} t - e^{t} + K)$ .

Tap for more steps...

Step 3.2.2.1.1

Apply the distributive property.

$x^{2} = 2 (e^{t} t) + 2 (- e^{t}) + 2 K$

Step 3.2.2.1.2

Multiply $- 1$ by $2$ .

$x^{2} = 2 e^{t} t - 2 e^{t} + 2 K$

Step 3.2.2.1.3

Reorder factors in $2 e^{t} t - 2 e^{t} + 2 K$ .

$x^{2} = 2 t e^{t} - 2 e^{t} + 2 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2 t e^{t} - 2 e^{t} + 2 K}$

Step 3.4

Factor $2$ out of $2 t e^{t} - 2 e^{t} + 2 K$ .

Tap for more steps...

Step 3.4.1

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

$x = \pm \sqrt{2 (t e^{t}) - 2 e^{t} + 2 K}$

Step 3.4.2

Factor $2$ out of $- 2 e^{t}$ .

$x = \pm \sqrt{2 (t e^{t}) + 2 (- e^{t}) + 2 K}$

Step 3.4.3

Factor $2$ out of $2 K$ .

$x = \pm \sqrt{2 (t e^{t}) + 2 (- e^{t}) + 2 K}$

Step 3.4.4

Factor $2$ out of $2 (t e^{t}) + 2 (- e^{t})$ .

$x = \pm \sqrt{2 (t e^{t} - e^{t}) + 2 K}$

Step 3.4.5

Factor $2$ out of $2 (t e^{t} - e^{t}) + 2 K$ .

$x = \pm \sqrt{2 (t e^{t} - e^{t} + K)}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2 (t e^{t} - e^{t} + K)}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2 (t e^{t} - e^{t} + K)}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2 (t e^{t} - e^{t} + K)}$

$x = - \sqrt{2 (t e^{t} - e^{t} + K)}$

$x = \sqrt{2 (t e^{t} - e^{t} + K)}$

$x = - \sqrt{2 (t e^{t} - e^{t} + K)}$

$x = \sqrt{2 (t e^{t} - e^{t} + K)}$

$x = - \sqrt{2 (t e^{t} - e^{t} + K)}$