Calculus Examples

$\frac{d y}{d x} = \sin (2 x + π)$ , $y (0) = 6$

Step 1

Rewrite the equation.

$d y = \sin (2 x + π) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \sin (2 x + π) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \sin (2 x + π) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1.1

Differentiate $2 x + π$ .

$\frac{d}{d x} [2 x + π]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $2 x + π$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [π]$ .

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

Step 2.3.1.1.3

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

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Step 2.3.1.1.3.1

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

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

Step 2.3.1.1.3.2

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

$2 \cdot 1 + \frac{d}{d x} [π]$

Step 2.3.1.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [π]$

Step 2.3.1.1.4

Differentiate using the Constant Rule.

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Step 2.3.1.1.4.1

Since $π$ is constant with respect to $x$ , the derivative of $π$ with respect to $x$ is $0$ .

$2 + 0$

Step 2.3.1.1.4.2

Add $2$ and $0$ .

$2$

Step 2.3.1.2

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

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

Step 2.3.5.2

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

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

Step 2.3.6

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

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

Step 2.3.7

Reorder terms.

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

Step 2.4

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

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

Step 3

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $6$ for $y$ in $y = - \frac{1}{2} \cos (2 x + π) + K$ .

$6 = - \frac{1}{2} \cos (2 \cdot 0 + π) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $- \frac{1}{2} \cdot \cos (2 \cdot 0 + π) + K = 6$ .

$- \frac{1}{2} \cdot \cos (2 \cdot 0 + π) + K = 6$

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

Multiply $2$ by $0$ .

$- \frac{1}{2} \cdot \cos (0 + π) + K = 6$

Step 4.2.1.2

Add $0$ and $π$ .

$- \frac{1}{2} \cdot \cos (π) + K = 6$

Step 4.2.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \frac{1}{2} \cdot (- \cos (0)) + K = 6$

Step 4.2.1.4

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

$- \frac{1}{2} \cdot (- 1 \cdot 1) + K = 6$

Step 4.2.1.5

Multiply $- 1$ by $1$ .

$- \frac{1}{2} \cdot - 1 + K = 6$

Step 4.2.1.6

Multiply $- \frac{1}{2} \cdot - 1$ .

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Step 4.2.1.6.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) + K = 6$

Step 4.2.1.6.2

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

$\frac{1}{2} + K = 6$

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

$K = 6 - \frac{1}{2}$

Step 4.3.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$K = 6 \cdot \frac{2}{2} - \frac{1}{2}$

Step 4.3.3

Combine $6$ and $\frac{2}{2}$ .

$K = \frac{6 \cdot 2}{2} - \frac{1}{2}$

Step 4.3.4

Combine the numerators over the common denominator.

$K = \frac{6 \cdot 2 - 1}{2}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $6$ by $2$ .

$K = \frac{12 - 1}{2}$

Step 4.3.5.2

Subtract $1$ from $12$ .

$K = \frac{11}{2}$

Step 5

Substitute $\frac{11}{2}$ for $K$ in $y = - \frac{1}{2} \cos (2 x + π) + K$ and simplify.

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Step 5.1

Substitute $\frac{11}{2}$ for $K$ .

$y = - \frac{1}{2} \cos (2 x + π) + \frac{11}{2}$

Step 5.2

Combine $\cos (2 x + π)$ and $\frac{1}{2}$ .

$y = - \frac{\cos (2 x + π)}{2} + \frac{11}{2}$