Calculus Examples

$\frac{d y}{d x} = \sin (x)$ , $y (0) = 5$

Step 1

Rewrite the equation.

$d y = \sin (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \sin (x) d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $5$ for $y$ in $y = - \cos (x) + K$ .

$5 = - \cos (0) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \cos (0) + K = 5$ .

$- \cos (0) + K = 5$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$- 1 \cdot 1 + K = 5$

Step 4.2.1.2

Multiply $- 1$ by $1$ .

$- 1 + K = 5$

Step 4.3

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

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Step 4.3.1

Add $1$ to both sides of the equation.

$K = 5 + 1$

Step 4.3.2

Add $5$ and $1$ .

$K = 6$

Step 5

Substitute $6$ for $K$ in $y = - \cos (x) + K$ and simplify.

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Step 5.1

Substitute $6$ for $K$ .

$y = - \cos (x) + 6$