RUMUS DASAR
| \(\int \sin x \: dx\) | \(- \cos x + c\) |
| \(\int \cos x \: dx\) | \(\sin x + c\) |
| \(\int \tan x \: dx\) | \(\ln | \sec x | + c\) |
| \(\int \cot x \: dx\) | \(\ln | \sin x | + c\) |
| \(\int \sec x \: dx\) | \(\ln | \sec x + \tan x | + c\) |
| \(\int \csc x \: dx\) | \(\ln | \csc x – \cot x | + c\) |
| \(\int \sec^2 x \: dx\) | \(\tan x + c\) |
| \(\int \csc^2 x \: dx\) | \(\cot x + c\) |
| \(\int \sec x \tan x \: dx\) | \(\sec x + c\) |
| \(\int \csc x \cot x \: dx\) | \(- \csc x + c\) |
Contoh 01
\begin{equation*} \begin{split} & \int \sin(3x+2)\:dx \\\\ & \int \sin(3x+2)\:\frac{d(3x+2)}{3} \\\\ & \frac{1}{3}\int \sin(3x+2)\:d(3x+2) \\\\ & \bbox[5px, border: 2px solid magenta] {-\frac{1}{3}\cos(3x+2)+ c} \end{split} \end{equation*}
Contoh 02
\begin{equation*} \begin{split} & \int \cos(1-2x)\:dx\\\\ & \int \cos(1-2x)\:\frac{d(1-2x)}{-2}\\\\ & -\frac{1}{2}\int \cos(1-2x)\:d(1-2x)\\\\ & \bbox[5px, border: 2px solid magenta] {-\frac{1}{2}\sin(1-2x)+ c} \end{split} \end{equation*}
Contoh 03
\begin{equation*} \begin{split} & \int \tan(2x + 1)\:dx\\\\ & \int \tan(2x + 1)\:\frac{d(2x + 1)}{2}\\\\ & \frac{1}{2}\int \tan(2x + 1)\:d(2x + 1)\\\\ & \bbox[5px, border: 2px solid magenta] {\frac{1}{2} \ln|\sec (2x + 1) | + c} \end{split} \end{equation*}
SOAL LATIHAN
[s2If current_user_can(access_s2member_level1)]
\(\int \sec^2(-4x + 1)\:dx\)
\(\int \sec^2(-4x + 1)\:dx\)
Pembahasan:
\begin{equation*} \begin{split} & \int \sec^2(-4x + 1)\:dx\\\\ & \int \sec^2(-4x + 1)\:\frac{d(-4x + 1)}{-4}\\\\ & -\frac{1}{4}\int \sec^2(-4x + 1)\:d(-4x + 1)\\\\ & \bbox[5px, border: 2px solid magenta] {-\frac{1}{4} \tan (-4x + 1)+ c} \end{split} \end{equation*}
\(\int \tan^2 4x\:dx\)
\(\int \tan^2 4x\:dx\)
Pembahasan:
\begin{equation*} \begin{split} & \int \tan^2 4x\:dx\\ & \int \tan^2 4x\:\frac{d(4x)}{4}\\\\ & \frac{1}{4}\int \tan^2 4x\:d(4x)\\\\ & \frac{1}{4}\int [\sec^2 4x – 1] \: d(4x)\\\\ & \frac{1}{4}\int \sec^2 4x \: d(4x) – \frac 14 \int \: d(4x)\\\\ & \frac 14 \tan 4x – \frac 14 \:.\: 4x + C \\\\ & \bbox[5px, border: 2px solid magenta] {\frac 14 \tan 4x – x + c} \end{split} \end{equation*}
\(\int \sec (5x + 3) \tan(5x + 3)\:dx\)
\(\int \sec (5x + 3) \tan(5x + 3)\:dx\)
Pembahasan:
\begin{equation*} \begin{split} & \int \sec (5x + 3) \tan(5x + 3)\:dx\\\\ & \int \sec (5x + 3) \tan(5x + 3)\:\frac{d(5x + 3)}{5}\\\\ & \frac{1}{5}\int \sec (5x + 3) \tan(5x + 3)\:d(5x + 3)\\\\ & \bbox[5px, border: 2px solid magenta] {\frac{1}{5} \sec (5x + 3)+ c} \end{split} \end{equation*}
\(\int x\tan x^2\:dx\)
\(\int x\tan x^2\:dx\)
Pembahasan:
\begin{equation*} \begin{split} & \int x\tan x^2\:dx\\\\ & \int \cancel {x}\tan x^2\:\frac{d(x^2)}{2\cancel {x}}\\\\ & \frac{1}{2}\int \tan x^2\:d(x^2)\\\\ & \bbox[5px, border: 2px solid magenta] {-\frac{1}{2}\ln |\sec x^2|+ c} \end{split} \end{equation*}
[else]
[/s2If]