Calculus Examples

$\frac{d s}{d t} = 5 t + \cos (t)$ , $s (0) = 0$

Step 1

Rewrite the equation.

$d s = (5 t + \cos (t)) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d s = \int 5 t + \cos (t) d t$

Step 2.2

Apply the constant rule.

$s + C_{1} = \int 5 t + \cos (t) d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$s + C_{1} = \int 5 t d t + \int \cos (t) d t$

Step 2.3.2

Since $5$ is constant with respect to $t$ , move $5$ out of the integral.

$s + C_{1} = 5 \int t d t + \int \cos (t) d t$

Step 2.3.3

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

$s + C_{1} = 5 (\frac{1}{2} t^{2} + C_{2}) + \int \cos (t) d t$

Step 2.3.4

The integral of $\cos (t)$ with respect to $t$ is $\sin (t)$ .

$s + C_{1} = 5 (\frac{1}{2} t^{2} + C_{2}) + \sin (t) + C_{3}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

$s + C_{1} = 5 (\frac{t^{2}}{2} + C_{2}) + \sin (t) + C_{3}$

Step 2.3.5.2

Simplify.

$s + C_{1} = \frac{5 t^{2}}{2} + \sin (t) + C_{4}$

Step 2.3.5.3

Reorder terms.

$s + C_{1} = \frac{5}{2} t^{2} + \sin (t) + C_{4}$

Step 2.4

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

$s = \frac{5}{2} t^{2} + \sin (t) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $t$ and $0$ for $s$ in $s = \frac{5}{2} t^{2} + \sin (t) + K$ .

$0 = \frac{5}{2} \cdot 0^{2} + \sin (0) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{5}{2} \cdot 0^{2} + \sin (0) + K = 0$ .

$\frac{5}{2} \cdot 0^{2} + \sin (0) + K = 0$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $\frac{5}{2} \cdot 0^{2} + \sin (0) + K$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{5}{2} \cdot 0 + \sin (0) + K = 0$

Step 4.2.1.1.2

Multiply $\frac{5}{2}$ by $0$ .

$0 + \sin (0) + K = 0$

Step 4.2.1.1.3

The exact value of $\sin (0)$ is $0$ .

$0 + 0 + K = 0$

Step 4.2.1.2

Combine the opposite terms in $0 + 0 + K$ .

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Step 4.2.1.2.1

Add $0$ and $0$ .

$0 + K = 0$

Step 4.2.1.2.2

Add $0$ and $K$ .

$K = 0$

Step 5

Substitute $0$ for $K$ in $s = \frac{5}{2} t^{2} + \sin (t) + K$ and simplify.

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Step 5.1

Substitute $0$ for $K$ .

$s = \frac{5}{2} t^{2} + \sin (t) + 0$

Step 5.2

Add $\frac{5}{2} t^{2} + \sin (t)$ and $0$ .

$s = \frac{5}{2} t^{2} + \sin (t)$

Step 5.3

Combine $\frac{5}{2}$ and $t^{2}$ .

$s = \frac{5 t^{2}}{2} + \sin (t)$