Calculus Examples

$\frac{d y}{d t} = t e^{t}$

Step 1

Rewrite the equation.

$d y = t e^{t} d t$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int t e^{t} d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int t e^{t} d t$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

$y + C_{1} = t e^{t} - \int e^{t} d t$

Step 2.3.2

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

$y + C_{1} = t e^{t} - (e^{t} + C_{2})$

Step 2.3.3

Simplify.

$y + C_{1} = t e^{t} - e^{t} + C_{2}$

Step 2.3.4

Reorder terms.

$y + C_{1} = e^{t} t - e^{t} + C_{2}$

Step 2.4

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

$y = e^{t} t - e^{t} + K$