Calculus Examples

$\int \tan (5 x) d x$

Step 1

Let

$u = 5 x$ . Then

$d u = 5 d x$ , so

$\frac{1}{5} d u = d x$ . Rewrite using

$u$ and

$d$

$u$ .

Tap for more steps...

Step 1.1

Let

$u = 5 x$ . Find

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

Tap for more steps...

Step 1.1.1

Differentiate

$5 x$ .

$\frac{d}{d x} [5 x]$

Step 1.1.2

Since

$5$ is constant with respect to

$x$ , the derivative of

$5 x$ with respect to

$x$ is

$5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$5 \cdot 1$

Step 1.1.4

Multiply

$5$ by

$1$ .

$5$

$5$

Step 1.2

Rewrite the problem using

$u$ and

$d u$ .

$\int \tan (u) \frac{1}{5} d u$

$\int \tan (u) \frac{1}{5} d u$

Step 2

Combine

$\tan (u)$ and

$\frac{1}{5}$ .

$\int \frac{\tan (u)}{5} d u$

Step 3

Since

$\frac{1}{5}$ is constant with respect to

$u$ , move

$\frac{1}{5}$ out of the integral.

$\frac{1}{5} \int \tan (u) d u$

Step 4

The integral of

$\tan (u)$ with respect to

$u$ is

$\ln (| \sec (u) |)$ .

$\frac{1}{5} (\ln (| \sec (u) |) + C)$

Step 5

Simplify.

$\frac{1}{5} \ln (| \sec (u) |) + C$

Step 6

Replace all occurrences of

$u$ with

$5 x$ .

$\frac{1}{5} \ln (| \sec (5 x) |) + C$

$\int_{}^{} \tan (5 x) d x$

(

(

)

)

|

|

[

[

]

]

√

√





≥

≥





∫

∫













7

7

8

8

9

9













≤

≤





°

°

θ

θ









4

4

5

5

6

6

/

/

^

^

×

×

>

>





π

π













1

1

2

2

3

3

-

-

+

+

÷

÷

<

<





∞

∞





!

!



,

,

0

0

.

.

%

%



=

=









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