Calculus Examples

$\frac{d s}{d t} = \cos (t) - \sin (t)$ , $s (0) = 3$

Step 1

Rewrite the equation.

$d s = (\cos (t) - \sin (t)) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d s = \int \cos (t) - \sin (t) d t$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

$s + C_{1} = \sin (t) + C_{2} + \int - \sin (t) d t$

Step 2.3.3

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$s + C_{1} = \sin (t) + C_{2} - \int \sin (t) d t$

Step 2.3.4

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

$s + C_{1} = \sin (t) + C_{2} - (- \cos (t) + C_{3})$

Step 2.3.5

Simplify.

$s + C_{1} = \sin (t) + \cos (t) + C_{4}$

Step 2.4

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

$s = \sin (t) + \cos (t) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $t$ and $3$ for $s$ in $s = \sin (t) + \cos (t) + K$ .

$3 = \sin (0) + \cos (0) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\sin (0) + \cos (0) + K = 3$ .

$\sin (0) + \cos (0) + K = 3$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $\sin (0) + \cos (0) + K$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

$0 + \cos (0) + K = 3$

Step 4.2.1.1.2

The exact value of $\cos (0)$ is $1$ .

$0 + 1 + K = 3$

Step 4.2.1.2

Add $0$ and $1$ .

$1 + K = 3$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Subtract $1$ from both sides of the equation.

$K = 3 - 1$

Step 4.3.2

Subtract $1$ from $3$ .

$K = 2$

Step 5

Substitute $2$ for $K$ in $s = \sin (t) + \cos (t) + K$ and simplify.

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Step 5.1

Substitute $2$ for $K$ .

$s = \sin (t) + \cos (t) + 2$