Calculus Examples

$\frac{d y}{d t} = \sin (t) + 1$ , $y (\frac{π}{3}) = \frac{1}{2}$

Step 1

Rewrite the equation.

$d y = (\sin (t) + 1) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \sin (t) + 1 d t$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \sin (t) + 1 d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int \sin (t) d t + \int d t$

Step 2.3.2

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

$y + C_{1} = - \cos (t) + C_{2} + \int d t$

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Simplify.

$y + C_{1} = - \cos (t) + t + C_{4}$

Step 2.3.5

Reorder terms.

$y + C_{1} = t - \cos (t) + C_{4}$

Step 2.4

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

$y = t - \cos (t) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $\frac{π}{3}$ for $t$ and $\frac{1}{2}$ for $y$ in $y = t - \cos (t) + K$ .

$\frac{1}{2} = \frac{π}{3} - \cos (\frac{π}{3}) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\frac{π}{3} - \cos (\frac{π}{3}) + K = \frac{1}{2}$ .

$\frac{π}{3} - \cos (\frac{π}{3}) + K = \frac{1}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{π}{3} - \frac{1}{2} + K = \frac{1}{2}$

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 $\frac{π}{3}$ from both sides of the equation.

$- \frac{1}{2} + K = \frac{1}{2} - \frac{π}{3}$

Step 4.3.2

Add $\frac{1}{2}$ to both sides of the equation.

$K = \frac{1}{2} - \frac{π}{3} + \frac{1}{2}$

Step 4.3.3

Combine the numerators over the common denominator.

$K = \frac{1 + 1}{2} + \frac{- π}{3}$

Step 4.3.4

Add $1$ and $1$ .

$K = \frac{2}{2} + \frac{- π}{3}$

Step 4.3.5

Simplify each term.

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Step 4.3.5.1

Divide $2$ by $2$ .

$K = 1 + \frac{- π}{3}$

Step 4.3.5.2

Move the negative in front of the fraction.

$K = 1 - \frac{π}{3}$

Step 5

Substitute $1 - \frac{π}{3}$ for $K$ in $y = t - \cos (t) + K$ and simplify.

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Step 5.1

Substitute $1 - \frac{π}{3}$ for $K$ .

$y = t - \cos (t) + 1 - \frac{π}{3}$