Calculus Examples

$\frac{d y}{d x} = \sin (x)$ , $y (\frac{π}{2}) = 1$

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 $\frac{π}{2}$ for $x$ and $1$ for $y$ in $y = - \cos (x) + K$ .

$1 = - \cos (\frac{π}{2}) + K$

Step 4

Solve for $K$ .

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Step 4.1

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

$- \cos (\frac{π}{2}) + K = 1$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $- \cos (\frac{π}{2}) + K$ .

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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 $\cos (\frac{π}{2})$ is $0$ .

$- 0 + K = 1$

Step 4.2.1.1.2

Multiply $- 1$ by $0$ .

$0 + K = 1$

Step 4.2.1.2

Add $0$ and $K$ .

$K = 1$

Step 5

Substitute $1$ for $K$ in $y = - \cos (x) + K$ and simplify.

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Step 5.1

Substitute $1$ for $K$ .

$y = - \cos (x) + 1$