Kalkulus – Integral Tak Tentu Fungsi Trigonometri

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.

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

Pengembangan Rumus

  1. \(\int \sin ax dx = -\frac{1}{a} \cos ax + c\)
  2. \(\int \cos ax dx = \frac{1}{a} \sin ax + c\)
  3. \(\int \tan x dx = - \ln| \cos x| + c\)
  4. \(\int \cot x dx = \ln | \sin x | + c\)
  5. \(\int \cos (ax+b)dx=\frac{1}{a} \sin (ax+b)+c\)
  6. \(\int \sin (ax+b)dx=-\frac{1}{a} \cos (ax+b)+c\)
  7. \(\int \sec^{2}(ax+b) dx=\frac{1}{a} \tan (ax+b)+c\)
  8. \(\int \tan (ax+b) \sec (ax+b) dx=\frac{1}{a} \sec (ax+b)+c\)
  9. \(\int \csc^{2}(ax+b) dx=- \frac{1}{a} \cot (ax+b)+c\)
  10. \(\int \cot (ax+b) \csc (ax+b)dx=- \frac{1}{a} \csc (ax+b)+c\)

Contoh Soal

  1. \(\int (2sinx+3)dx\)= ...

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 :

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

Materi Lengkap

Berikut adalah materi lainnya yang membahas mengenai Integral.


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