Pembahasan Integral SBMPTN 2017 Matematika Dasar kode 224

Soal yang Akan Dibahas
$ \int \frac{1-x}{\sqrt{x}} dx = .... $
A). $ \frac{3}{2}(3+x)\sqrt{x} + C \, $
B). $ \frac{2}{3}(3-x)\sqrt{x} + C \, $
C). $ \frac{2}{3}(3+\sqrt{x})x + C \, $
D). $ \frac{1}{3\sqrt{x}} \left( \frac{1}{x} - 1 \right) + C \, $
E). $ \frac{1}{2\sqrt{x}} \left( \frac{1}{x} + 1 \right) + C $

$\spadesuit $ Konsep Dasar
*). Rumus integral aljabar :
$ \int x^n dx = \frac{1}{n+1}x^{n+1} + c $
*). Sifat-sifat Eksponen :
$ \frac{a^m}{a^n} = a^{m-n} $
$ a^{m+n} = a^m.a^n $
$ \frac{1}{a^n} = a^{-n} $
$ \sqrt{a} = a^\frac{1}{2} $

$\clubsuit $ Pembahasan
*). Menyelesaikan soal :
$\begin{align} \int \frac{1-x}{\sqrt{x}} dx & = \int \frac{1-x}{x^\frac{1}{2}} dx \\ & = \int \left( \frac{1}{x^\frac{1}{2}} - \frac{x}{x^\frac{1}{2}} \right) dx \\ & = \int \left( x^{-\frac{1}{2}} - x^\frac{1}{2} \right) dx \\ & = \frac{1}{-\frac{1}{2} + 1} x^{-\frac{1}{2} + 1} - \frac{1}{\frac{1}{2} + 1}x^{\frac{1}{2} + 1} + c \\ & = \frac{1}{\frac{1}{2}} x^{\frac{1}{2} } - \frac{1}{\frac{3}{2}}x^{\frac{1}{2} + 1} + c \\ & = 2 \sqrt{x} - \frac{2}{3}\sqrt{x}.x + c \\ & = \frac{2}{3}\sqrt{x} (3 - x) + c \\ & = \frac{2}{3}(3 - x)\sqrt{x} + c \end{align} $
Jadi, hasil integralnya adalah $ \frac{2}{3}(3 - x)\sqrt{x} + c . \, \heartsuit $

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