Calculus Examples

$\int \cos (3 t) d t$

Step 1

Let

$u = 3 t$ . Then

$d u = 3 d t$ , so

$\frac{1}{3} d u = d t$ . Rewrite using

$u$ and

$d$

$u$ .

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Step 1.1

Let

$u = 3 t$ . Find

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

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Step 1.1.1

Differentiate

$3 t$ .

$\frac{d}{d t} [3 t]$

Step 1.1.2

Since

$3$ is constant with respect to

$t$ , the derivative of

$3 t$ with respect to

$t$ is

$3 \frac{d}{d t} [t]$ .

$3 \frac{d}{d t} [t]$

Step 1.1.3

Differentiate using the Power Rule which states that

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

$n t^{n - 1}$ where

$n = 1$ .

$3 \cdot 1$

Step 1.1.4

Multiply

$3$ by

$1$ .

$3$

$3$

Step 1.2

Rewrite the problem using

$u$ and

$d u$ .

$\int \cos (u) \frac{1}{3} d u$

$\int \cos (u) \frac{1}{3} d u$

Step 2

Combine

$\cos (u)$ and

$\frac{1}{3}$ .

$\int \frac{\cos (u)}{3} d u$

Step 3

Since

$\frac{1}{3}$ is constant with respect to

$u$ , move

$\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \cos (u) d u$

Step 4

The integral of

$\cos (u)$ with respect to

$u$ is

$\sin (u)$ .

$\frac{1}{3} (\sin (u) + C)$

Step 5

Simplify.

$\frac{1}{3} \sin (u) + C$

Step 6

Replace all occurrences of

$u$ with

$3 t$ .

$\frac{1}{3} \sin (3 t) + C$

Enter a problem...

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