Calculus Examples

$\frac{d y}{d x} = 4 \cos (x - 1)$ , $y (1) = 6$

Step 1

Rewrite the equation.

$d y = 4 \cos (x - 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 4 \cos (x - 1) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 4 \cos (x - 1) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$y + C_{1} = 4 \int \cos (x - 1) d x$

Step 2.3.2

Let $u = x - 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = x - 1$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 2.3.2.1.3

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

$1 + \frac{d}{d x} [- 1]$

Step 2.3.2.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.2.1.5

Add $1$ and $0$ .

$1$

Step 2.3.2.2

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

$y + C_{1} = 4 \int \cos (u) d u$

Step 2.3.3

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

$y + C_{1} = 4 (\sin (u) + C_{2})$

Step 2.3.4

Simplify.

$y + C_{1} = 4 \sin (u) + C_{2}$

Step 2.3.5

Replace all occurrences of $u$ with $x - 1$ .

$y + C_{1} = 4 \sin (x - 1) + C_{2}$

Step 2.4

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

$y = 4 \sin (x - 1) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $1$ for $x$ and $6$ for $y$ in $y = 4 \sin (x - 1) + K$ .

$6 = 4 \sin (1 - 1) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $4 \sin (1 - 1) + K = 6$ .

$4 \sin (1 - 1) + K = 6$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $4 \sin (1 - 1) + K$ .

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Step 4.2.1.1

Subtract $1$ from $1$ .

$4 \sin (0) + K = 6$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

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

$4 \cdot 0 + K = 6$

Step 4.2.1.2.2

Multiply $4$ by $0$ .

$0 + K = 6$

Step 4.2.1.3

Add $0$ and $K$ .

$K = 6$

Step 5

Substitute $6$ for $K$ in $y = 4 \sin (x - 1) + K$ and simplify.

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Step 5.1

Substitute $6$ for $K$ .

$y = 4 \sin (x - 1) + 6$