selesaikan integral berikut!

Berikut ini adalah pertanyaan dari dwialfian767 pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas

Selesaikan integral berikut!

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

$\displaystyle\int\frac{x+2-\sqrt{x}}{\sqrt[3]{x}}\,dx$

$= \large\text{$\displaystyle\bf\frac{3}{5}x^{(5/3)}\:+\:3x^{(2/3)}\:-\:\frac{6}{7}x^{(7/6)}\:+\:C$}$

$= \large\text{$\displaystyle\bf\frac{3}{5}x\sqrt[\bf3]{\bf x^2}\:+\:3\sqrt[\bf3]{\bf x^2}\:-\:\frac{6}{7}x\sqrt[\bf6]{\bf x}\:+\:C$}$

$= \large\text{$\displaystyle\bf\frac{3x^2}{5\sqrt[\bf3]{\bf x}}\:+\:3\sqrt[\bf3]{\bf x^2}\:-\:\frac{6}{7}x\sqrt[\bf6]{\bf x}\:+\:C$}$

Pembahasan

Integral Tak Tentu

$\begin{aligned}&\int\frac{x+2-\sqrt{x}}{\sqrt[3]{x}}\,dx\\{=\ }&\int\left(\frac{x}{\sqrt[3]{x}}+\frac{2}{\sqrt[3]{x}}-\frac{\sqrt{x}}{\sqrt[3]{x}}\right)dx\\{=\ }&\int\left(x^{(1\:-\:1/3)}+2x^{(-1/3)}-x^{(1/2\:-1/3)}\right)dx\\{=\ }&\int\left(x^{(2/3)}+2x^{(-1/3)}-x^{(1/6)}\right)dx\\{=\ }&\int x^{(2/3)}dx\:+\:2\cdot\!\!\int x^{(-1/3)}dx\:-\:\int x^{(1/6)}dx\end{aligned}$
$\begin{aligned}{=\ }&\frac{x^{(2/3\:+\:1)}}{2/3\:+\:1}\:+\:2\cdot\frac{x^{(-1/3\:+\:1)}}{-1/3\:+\:1}\:-\:\frac{x^{(1/6\:+\:1)}}{1/6\:+\:1}\:+\:C\\{=\ }&\frac{x^{(5/3)}}{5/3}\:+\:2\cdot\frac{x^{(2/3)}}{2/3}\:-\:\frac{x^{(7/6)}}{7/6}\:+\:C\\{=\ }&\frac{3x^{(5/3)}}{5}\:+\:2\cdot\frac{3x^{(2/3)}}{2}\:-\:\frac{6x^{(7/6)}}{7}\:+\:C\end{aligned}$
$\begin{aligned}{=\ }&\large\text{$\boxed{\ \bf\frac{3}{5}x^{(5/3)}\:+\:3x^{(2/3)}\:-\:\frac{6}{7}x^{(7/6)}\:+\:C\ }$}\\{=\ }&\large\text{$\boxed{\ \bf\frac{3}{5}x\sqrt[\bf3]{\bf x^2}\:+\:3\sqrt[\bf3]{\bf x^2}\:-\:\frac{6}{7}x\sqrt[\bf6]{\bf x}\:+\:C\ }$}\\{=\ }&\large\text{$\boxed{\ \bf\frac{3x^2}{5\sqrt[\bf3]{\bf x}}\:+\:3\sqrt[\bf3]{\bf x^2}\:-\:\frac{6}{7}x\sqrt[\bf6]{\bf x}\:+\:C\ }$}\\\end{aligned}$
$\blacksquare$