Calculus Examples

\int 4 cos (2 x) d x

$\int 4 \cos (2 x) d x$

Step 1

Since

4

$4$ is constant with respect to

x

$x$ , move

4

$4$ out of the integral.

4 \int cos (2 x) d x

$4 \int \cos (2 x) d x$

Step 2

Let

u = 2 x

$u = 2 x$ . Then

d u = 2 d x

$d u = 2 d x$ , so

\frac{1}{2} d u = d x

$\frac{1}{2} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 2.1

Let

u = 2 x

$u = 2 x$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate

2 x

$2 x$ .

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

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

Step 2.1.2

Since

2

$2$ is constant with respect to

x

$x$ , the derivative of

2 x

$2 x$ with respect to

x

$x$ is

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

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

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

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

Step 2.1.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

2 \cdot 1

$2 \cdot 1$

Step 2.1.4

Multiply

2

$2$ by

1

$1$ .

2

$2$

2

$2$

Step 2.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

4 \int cos (u) \frac{1}{2} d u

$4 \int \cos (u) \frac{1}{2} d u$

4 \int cos (u) \frac{1}{2} d u

$4 \int \cos (u) \frac{1}{2} d u$

Step 3

Combine

cos (u)

$\cos (u)$ and

\frac{1}{2}

$\frac{1}{2}$ .

4 \int \frac{cos (u)}{2} d u

$4 \int \frac{\cos (u)}{2} d u$

Step 4

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

u

$u$ , move

\frac{1}{2}

$\frac{1}{2}$ out of the integral.

4 (\frac{1}{2} \int cos (u) d u)

$4 (\frac{1}{2} \int \cos (u) d u)$

Step 5

Simplify.

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Step 5.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

4

$4$ .

\frac{4}{2} \int cos (u) d u

$\frac{4}{2} \int \cos (u) d u$

Step 5.2

Cancel the common factor of

4

$4$ and

2

$2$ .

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Step 5.2.1

Factor

2

$2$ out of

4

$4$ .

\frac{2 \cdot 2}{2} \int cos (u) d u

$\frac{2 \cdot 2}{2} \int \cos (u) d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor

$2$ out of

$2$ .

$\frac{2 \cdot 2}{2 (1)} \int \cos (u) d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} \int \cos (u) d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{2}{1} \int \cos (u) d u$

Step 5.2.2.4

Divide

$2$ by

$1$ .

$2 \int \cos (u) d u$

$2 \int \cos (u) d u$

$2 \int \cos (u) d u$

$2 \int \cos (u) d u$

Step 6

The integral of

$\cos (u)$ with respect to

$u$ is

$\sin (u)$ .

$2 (\sin (u) + C)$

Step 7

Simplify.

$2 \sin (u) + C$

Step 8

Replace all occurrences of

$u$ with

$2 x$ .

$2 \sin (2 x) + C$

$\int_{}^{} 4 \cos (2 x) d x$

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$