Kalkulus – Integral Tak Tentu Fungsi Trigonometri

📋 Daftar Isi

Rumus Dasar

\[ \int sin x dx=-cos x +c \] \[ \int cos x dx= sin x +c \]

atau dapat dilihat dari tabel turunan fungsi trigonometri berikut.

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\[f(x)\]	\[f'(x)\]
\[\sin x\]	\[\cos x\]
\[\cos x\]	\[-\sin x\]
\[\tan x\]	\[\sec^{2}x\]
\[\sec x\]	\[\tan x \sec x\]
\[\cot x\]	\[-\csc^{2}x\]
\[\csc x\]	\[-\cot x \csc x\]

Pengembangan Rumus

\(\int \sin ax dx = -\frac{1}{a} \cos ax + c\)
\(\int \cos ax dx = \frac{1}{a} \sin ax + c\)
\(\int \tan x dx = – \ln| \cos x| + c\)
\(\int \cot x dx = \ln | \sin x | + c\)
\(\int \cos (ax+b)dx=\frac{1}{a} \sin (ax+b)+c\)
\(\int \sin (ax+b)dx=-\frac{1}{a} \cos (ax+b)+c\)
\(\int \sec^{2}(ax+b) dx=\frac{1}{a} \tan (ax+b)+c\)
\(\int \tan (ax+b) \sec (ax+b) dx=\frac{1}{a} \sec (ax+b)+c\)
\(\int \csc^{2}(ax+b) dx=- \frac{1}{a} \cot (ax+b)+c\)
\(\int \cot (ax+b) \csc (ax+b)dx=- \frac{1}{a} \csc (ax+b)+c\)

Contoh Soal

\(\int (2sinx+3)dx\)= …

Jawab :

\[ \int (2sinx+3)dx=2\int sinx dx+\int 3dx=-2cosx+3x+c \]

2. \(\int (sec^{2}2x-1)dx\) = …

Jawab :

\[ \int (sec^{2}2x-1)dx=\int sec^{2}2xdx-\int dx=\frac{1}{2}tan2x-x+c \]

3. \(\int sin4xcos2xdx\) = …

Jawab :

\[\begin{aligned} \int sin4xcos2xdx &=\int \frac{1}{2}(sin6x+sin2x)dx\\ &=\frac{1}{2}\int (sin6x+sin2x)dx\\ &=\frac{1}{2}(-\frac{1}{6}cos6x-\frac{1}{2}cos2x+c\\ &=-\frac{1}{12}cos6x-\frac{1}{4}cos2x+c \end{aligned}\]