Calculus Examples

\int cos (2 t) d t

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

Step 1

Let

u = 2 t

$u = 2 t$ . Then

d u = 2 d t

$d u = 2 d t$ , so

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

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

u

$u$ and

d

$d$

u

$u$ .

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Step 1.1

Let

u = 2 t

$u = 2 t$ . Find

\frac{d u}{d t}

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

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Step 1.1.1

Differentiate

2 t

$2 t$ .

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

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

Step 1.1.2

Since

2

$2$ is constant with respect to

t

$t$ , the derivative of

2 t

$2 t$ with respect to

t

$t$ is

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

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

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

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

Step 1.1.3

Differentiate using the Power Rule which states that

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

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

n t^{n - 1}

$n t^{n - 1}$ where

n = 1

$n = 1$ .

2 \cdot 1

$2 \cdot 1$

Step 1.1.4

Multiply

2

$2$ by

1

$1$ .

2

$2$

2

$2$

Step 1.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

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

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

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

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

Step 2

Combine

$\cos (u)$ and

$\frac{1}{2}$ .

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

Step 3

Since

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

$u$ , move

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

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

Step 4

The integral of

$\cos (u)$ with respect to

$u$ is

$\sin (u)$ .

$\frac{1}{2} (\sin (u) + C)$

Step 5

Simplify.

$\frac{1}{2} \sin (u) + C$

Step 6

Replace all occurrences of

$u$ with

$2 t$ .

$\frac{1}{2} \sin (2 t) + C$

$\int_{}^{} \cos (2 t) d t$

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