Calculus Examples

$C (t) = 3 t e^{- \frac{t}{3}}$

Step 1

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$3 \int t e^{- \frac{t}{3}} d t$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = t$ and $d v = e^{- \frac{t}{3}}$ .

$3 (t (- 3 e^{- \frac{1}{3} t}) - \int - 3 e^{- \frac{1}{3} t} d t)$

Step 3

Since $- 3$ is constant with respect to $t$ , move $- 3$ out of the integral.

$3 (t (- 3 e^{- \frac{1}{3} t}) - (- 3 \int e^{- \frac{1}{3} t} d t))$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Combine $t$ and $\frac{1}{3}$ .

$3 (t (- 3 e^{- \frac{t}{3}}) - (- 3 \int e^{- \frac{1}{3} t} d t))$

Step 4.2

Combine $t$ and $\frac{1}{3}$ .

$3 (t (- 3 e^{- \frac{t}{3}}) - (- 3 \int e^{- \frac{t}{3}} d t))$

Step 4.3

Multiply $- 3$ by $- 1$ .

$3 (t (- 3 e^{- \frac{t}{3}}) + 3 \int e^{- \frac{t}{3}} d t)$

Step 5

Let $u = - \frac{t}{3}$ . Then $d u = - \frac{1}{3} d t$ , so $- 3 d u = d t$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.1

Let $u = - \frac{t}{3}$ . Find $\frac{d u}{d t}$ .

Tap for more steps...

Step 5.1.1

Differentiate $- \frac{t}{3}$ .

$\frac{d}{d t} [- \frac{t}{3}]$

Step 5.1.2

Since $- \frac{1}{3}$ is constant with respect to $t$ , the derivative of $- \frac{t}{3}$ with respect to $t$ is $- \frac{1}{3} \frac{d}{d t} [t]$ .

$- \frac{1}{3} \frac{d}{d t} [t]$

Step 5.1.3

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

$- \frac{1}{3} \cdot 1$

Step 5.1.4

Multiply $- 1$ by $1$ .

$- \frac{1}{3}$

Step 5.2

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

$3 (t (- 3 e^{- \frac{t}{3}}) + 3 \int - e^{u} \frac{- 1}{- \frac{1}{3}} d u)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Dividing two negative values results in a positive value.

$3 (t (- 3 e^{- \frac{t}{3}}) + 3 \int - e^{u} \frac{1}{\frac{1}{3}} d u)$

Step 6.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$3 (t (- 3 e^{- \frac{t}{3}}) + 3 \int - e^{u} (1 \cdot 3) d u)$

Step 6.3

Multiply $3$ by $1$ .

$3 (t (- 3 e^{- \frac{t}{3}}) + 3 \int - e^{u} \cdot 3 d u)$

Step 6.4

Multiply $3$ by $- 1$ .

$3 (t (- 3 e^{- \frac{t}{3}}) + 3 \int - 3 e^{u} d u)$

Step 7

Since $- 3$ is constant with respect to $u$ , move $- 3$ out of the integral.

$3 (t (- 3 e^{- \frac{t}{3}}) + 3 (- 3 \int e^{u} d u))$

Step 8

Multiply $- 3$ by $3$ .

$3 (t (- 3 e^{- \frac{t}{3}}) - 9 \int e^{u} d u)$

Step 9

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

$3 (t (- 3 e^{- \frac{t}{3}}) - 9 (e^{u} + C))$

Step 10

Rewrite $3 (t (- 3 e^{- \frac{t}{3}}) - 9 (e^{u} + C))$ as $3 (- 3 t e^{- \frac{t}{3}} - 9 e^{u}) + C$ .

$3 (- 3 t e^{- \frac{t}{3}} - 9 e^{u}) + C$

Step 11

Replace all occurrences of $u$ with $- \frac{t}{3}$ .

$3 (- 3 t e^{- \frac{t}{3}} - 9 e^{- \frac{t}{3}}) + C$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Apply the distributive property.

$3 (- 3 t e^{- \frac{t}{3}}) + 3 (- 9 e^{- \frac{t}{3}}) + C$

Step 12.2

Multiply $- 3$ by $3$ .

$- 9 (t e^{- \frac{t}{3}}) + 3 (- 9 e^{- \frac{t}{3}}) + C$

Step 12.3

Multiply $- 9$ by $3$ .

$- 9 t e^{- \frac{t}{3}} - 27 e^{- \frac{t}{3}} + C$

Step 13

Reorder terms.

$- 9 t e^{- \frac{1}{3} t} - 27 e^{- \frac{1}{3} t} + C$