Calculus Examples

$\frac{d E}{d t} = 25 - 8 t$

Step 1

Rewrite the equation.

$d E = (25 - 8 t) d t$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d E = \int 25 - 8 t d t$

Step 2.2

Apply the constant rule.

$E + C_{1} = \int 25 - 8 t d t$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Split the single integral into multiple integrals.

$E + C_{1} = \int 25 d t + \int - 8 t d t$

Step 2.3.2

Apply the constant rule.

$E + C_{1} = 25 t + C_{2} + \int - 8 t d t$

Step 2.3.3

Since $- 8$ is constant with respect to $t$ , move $- 8$ out of the integral.

$E + C_{1} = 25 t + C_{2} - 8 \int t d t$

Step 2.3.4

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

$E + C_{1} = 25 t + C_{2} - 8 (\frac{1}{2} t^{2} + C_{3})$

Step 2.3.5

Simplify.

Tap for more steps...

Step 2.3.5.1

Simplify.

$E + C_{1} = 25 t - 8 (\frac{1}{2}) t^{2} + C_{4}$

Step 2.3.5.2

Simplify.

Tap for more steps...

Step 2.3.5.2.1

Combine $- 8$ and $\frac{1}{2}$ .

$E + C_{1} = 25 t + \frac{- 8}{2} t^{2} + C_{4}$

Step 2.3.5.2.2

Cancel the common factor of $- 8$ and $2$ .

Tap for more steps...

Step 2.3.5.2.2.1

Factor $2$ out of $- 8$ .

$E + C_{1} = 25 t + \frac{2 \cdot - 4}{2} t^{2} + C_{4}$

Step 2.3.5.2.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.5.2.2.2.1

Factor $2$ out of $2$ .

$E + C_{1} = 25 t + \frac{2 \cdot - 4}{2 (1)} t^{2} + C_{4}$

Step 2.3.5.2.2.2.2

Cancel the common factor.

$E + C_{1} = 25 t + \frac{2 \cdot - 4}{2 \cdot 1} t^{2} + C_{4}$

Step 2.3.5.2.2.2.3

Rewrite the expression.

$E + C_{1} = 25 t + \frac{- 4}{1} t^{2} + C_{4}$

Step 2.3.5.2.2.2.4

Divide $- 4$ by $1$ .

$E + C_{1} = 25 t - 4 t^{2} + C_{4}$

Step 2.3.6

Reorder terms.

$E + C_{1} = - 4 t^{2} + 25 t + C_{4}$

Step 2.4

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

$E = - 4 t^{2} + 25 t + K$