3.1 Pengamiran Sebagai Songsangan Pembezaan

3.1 Pengamiran Sebagai Songsangan Pembezaan

1. Pengamiran ialah process songsangan pembezaan.

Jika \frac{d y}{d x} = f (x), maka \int f (x) d x = y .

(A) Pengamiran Pemalar

\int a d x = a x + c

Contoh:

\int 2 d x = 2 x + c

(B) Pengamiran axⁿ

\int a x^{n} d x = \frac{a x^{n + 1}}{n + 1} + c

Contoh 1:

\int 2 x^{3} d x = \frac{2 x^{4}}{4} + c = \frac{x^{4}}{2} + c

Contoh 2:

\begin{array}{l} \int \frac{2}{3 x^{5}} d x = \int \frac{2}{3} x^{- 5} d x \\ = \frac{2}{3} (\frac{x^{- 4}}{- 4}) + c \\ = \frac{2}{3} (\frac{x^{- 4}}{- 4}) + c \\ = \frac{x^{- 4}}{- 6} + c \end{array}

(C) Pengamiran Fungsi berbentuk Hasil Tambah Sebutan Algebra

\begin{array}{l} \int (u \pm v) d x = \int u d x \pm \int v d x \\ u dan v adalah fungsi dalam x \end{array}

Contoh 1:

\begin{array}{l} \int 3 x^{2} + 2 x d x \\ = \int 3 x^{2} d x + \int 2 x d x \\ = \frac{3 x^{3}}{3} + \frac{2 x^{2}}{2} + c \\ = \frac{3 x^{3}}{3} + \frac{2 x^{2}}{2} + c \\ = x^{3} + x^{2} + c \end{array}

Contoh 2:

\begin{array}{l} \int (x + 2) (3 x + 1) d x \\ = \int 3 x^{2} + 7 x + 2 d x \\ = \int 3 x^{2} d x + \int 7 x d x + \int 2 d x \\ = \frac{3 x^{3}}{3} + \frac{7 x^{2}}{2} + 2 x + c \\ = x^{3} + \frac{7 x^{2}}{2} + 2 x + c \end{array}

Contoh 3:

\begin{array}{l} \int \frac{3 x^{3} + x^{2} - x}{x} d x \\ = \int (3 x^{2} + x - 1) d x \\ = \int 3 x^{2} d x + \int x d x - \int 1 d x \\ = \frac{3 x^{3}}{3} + \frac{x^{2}}{2} - x + c \\ = x^{3} + \frac{x^{2}}{2} - x + c \end{array}