Calculus Examples

$\frac{d y}{d x} = \cos (2 x)$ , $y (\frac{π}{2}) = 3$

Step 1

Rewrite the equation.

$d y = \cos (2 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \cos (2 x) d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.3.1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.3.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 2.3.1.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.1.2

Rewrite the problem using $u$ and $d u$ .

$y + C_{1} = \int \cos (u) \frac{1}{2} d u$

Step 2.3.2

Combine $\cos (u)$ and $\frac{1}{2}$ .

$y + C_{1} = \int \frac{\cos (u)}{2} d u$

Step 2.3.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$y + C_{1} = \frac{1}{2} \int \cos (u) d u$

Step 2.3.4

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

$y + C_{1} = \frac{1}{2} (\sin (u) + C_{2})$

Step 2.3.5

Simplify.

$y + C_{1} = \frac{1}{2} \sin (u) + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $2 x$ .

$y + C_{1} = \frac{1}{2} \sin (2 x) + C_{2}$

Step 2.4

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

$y = \frac{1}{2} \sin (2 x) + K$

Step 3

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

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

Step 4

Solve for $K$ .

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Step 4.1

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

$\frac{1}{2} \cdot \sin (2 (\frac{π}{2})) + K = 3$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $\frac{1}{2} \cdot \sin (2 (\frac{π}{2})) + K$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1.1

Cancel the common factor.

$\frac{1}{2} \cdot \sin (2 \frac{π}{2}) + K = 3$

Step 4.2.1.1.1.2

Rewrite the expression.

$\frac{1}{2} \cdot \sin (π) + K = 3$

Step 4.2.1.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{1}{2} \cdot \sin (0) + K = 3$

Step 4.2.1.1.3

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

$\frac{1}{2} \cdot 0 + K = 3$

Step 4.2.1.1.4

Multiply $\frac{1}{2}$ by $0$ .

$0 + K = 3$

Step 4.2.1.2

Add $0$ and $K$ .

$K = 3$

Step 5

Substitute $3$ for $K$ in $y = \frac{1}{2} \sin (2 x) + K$ and simplify.

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Step 5.1

Substitute $3$ for $K$ .

$y = \frac{1}{2} \sin (2 x) + 3$

Step 5.2

Combine $\frac{1}{2}$ and $\sin (2 x)$ .

$y = \frac{\sin (2 x)}{2} + 3$