Pembahasan Integral UM UNDIP 2016 Matematika Dasar Ipa

Soal yang Akan Dibahas
$ \int x^5 \left( 2 - x^3 \right) ^\frac{1}{2} \, dx = .... $
A). $ \frac{2}{45}(3x^3+4)(-x^3+2)^\frac{3}{2} + c \, $
B). $ \frac{-2}{5}(3x^3+4)(-x^3+2)^\frac{3}{2} + c \, $
C). $ \frac{2}{5}(3x^3+4)(-x^3+2)^\frac{3}{2} + c \, $
D). $ \frac{-2}{25}(3x^3+4)(-x^3+2)^\frac{3}{2} + c \, $
E). $ \frac{-2}{45}(3x^3+4)(-x^3+2)^\frac{3}{2} + c \, $

$\spadesuit $ Konsep Dasar Integral :
*). Rumus umus integral :
$ \int au^n du = \frac{a}{n+1}u^{n+1} + c $
*). Salah satu metode penyelesaian integral adalah metode substitusi.

$\clubsuit $ Pembahasan
*). Misalkan $ u = 2 - x^3 \rightarrow x^3 = 2 - u $ :
$ \frac{du}{dx} = -3x^2 \rightarrow dx = \frac{du}{-3x^2 }$
*). Menyelesaikan soalnya dengan substitusi bentu yang ada :
$ \begin{align} & \int x^5 \left( 2 - x^3 \right) ^\frac{1}{2} \, dx \\ & = \int x^5 u^\frac{1}{2} \, \frac{du}{-3x^2 } \\ & = -\frac{1}{3}\int x^3 u^\frac{1}{2} \, du \\ & = -\frac{1}{3}\int (2 - u) u^\frac{1}{2} \, du \\ & = -\frac{1}{3}\int 2u^\frac{1}{2} - u^\frac{3}{2} \, du \\ & = -\frac{1}{3} \, \left( \frac{2}{\frac{1}{2}+1}u^{\frac{1}{2}+1} - \frac{1}{\frac{3}{2}+1}u^{\frac{3}{2}+1} \right) + c \\ & = -\frac{1}{3} \, \left( \frac{2}{\frac{3}{2} }u^{\frac{3}{2}} - \frac{1}{\frac{5}{2}}u^{\frac{5}{2}} \right) + c \\ & = -\frac{1}{3} \, \left( \frac{4}{3} u^{\frac{3}{2}} - \frac{2}{5}u^{\frac{5}{2}} \right) + c \\ & = -\frac{1}{3} \, \left( \frac{4}{3} - \frac{2}{5}u \right)u^{\frac{3}{2}} + c \\ & = -\frac{1}{3} \, \left( \frac{20}{15} - \frac{6}{15}u \right)u^{\frac{3}{2}} + c \\ & = -\frac{1}{3} \, \left( \frac{20}{15} - \frac{6}{15}(2-x^3) \right)(2-x^3)^{\frac{3}{2}} + c \\ & = -\frac{1}{3} \, \left( \frac{20}{15} - \frac{12}{15} + \frac{6}{15}x^3 \right)(2-x^3)^{\frac{3}{2}} + c \\ & = -\frac{1}{3} \, \left( \frac{8}{15} + \frac{6}{15}x^3 \right)(2-x^3)^{\frac{3}{2}} + c \\ & = -\frac{1}{3} . \frac{2}{15} \left( 4 + 3x^3 \right)(2-x^3)^{\frac{3}{2}} + c \\ & = -\frac{2}{45} \left( 3x^3 + 4 \right)(-x^3 + 2)^{\frac{3}{2}} + c \end{align} $
Jadi, hasilnya $ -\frac{2}{45} \left( 3x^3 + 4 \right)(-x^3 + 2)^{\frac{3}{2}} + c . \, \heartsuit $

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