Calculus Examples

$\frac{d y}{d x} = \cos (4 x) - 2 x$

Step 1

Rewrite the equation.

$d y = (\cos (4 x) - 2 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \cos (4 x) - 2 x d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \cos (4 x) - 2 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int \cos (4 x) d x + \int - 2 x d x$

Step 2.3.2

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 2.3.2.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

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 $n = 1$ .

$4 \cdot 1$

Step 2.3.2.1.4

Multiply $4$ by $1$ .

$4$

Step 2.3.2.2

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

$y + C_{1} = \int \cos (u) \frac{1}{4} d u + \int - 2 x d x$

Step 2.3.3

Combine $\cos (u)$ and $\frac{1}{4}$ .

$y + C_{1} = \int \frac{\cos (u)}{4} d u + \int - 2 x d x$

Step 2.3.4

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$y + C_{1} = \frac{1}{4} \int \cos (u) d u + \int - 2 x d x$

Step 2.3.5

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

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

Step 2.3.6

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 2.3.7

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$y + C_{1} = \frac{1}{4} (\sin (u) + C_{2}) - 2 (\frac{1}{2} x^{2} + C_{3})$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Simplify.

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

Step 2.3.8.2

Simplify.

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Step 2.3.8.2.1

Combine $- 2$ and $\frac{1}{2}$ .

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

Step 2.3.8.2.2

Cancel the common factor of $- 2$ and $2$ .

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Step 2.3.8.2.2.1

Factor $2$ out of $- 2$ .

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

Step 2.3.8.2.2.2

Cancel the common factors.

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Step 2.3.8.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.8.2.2.2.2

Cancel the common factor.

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

Step 2.3.8.2.2.2.3

Rewrite the expression.

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

Step 2.3.8.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 2.3.9

Replace all occurrences of $u$ with $4 x$ .

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

Step 2.3.10

Reorder terms.

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

Step 2.4

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

$y = - x^{2} + \frac{1}{4} \sin (4 x) + K$