1. Integral Substitusi
Ada beberapa bentuk integral yang terkadang pengintegralannya membutuhkan teknik tertentu. Di antara bentuk tertentu itu adalah dengan substitusi, yaitu:
$\displaystyle \int u^{n}.u'\: dx=\displaystyle \int u^{n}\: du=\displaystyle \frac{1}{n+1}u^{n+1}+C$ .
$\LARGE\colorbox{yellow}{CONTOH SOAL}$.
$\begin{array}{ll}\\ 1.&\textrm{Selesaikan integral berikut}\\ &\displaystyle \int \left ( x^{2}+3 \right )^{20}2x\: dx\\\\ &\textbf{Jawab}:\\ &\textrm{Misalkan}\\ &\left\{\begin{matrix} u & = & x^{2} &+&3&,&\textrm{maka} \\ du &= &2x & dx \end{matrix}\right.\\ &\textrm{sehingga dengan integral substitusi}\\ &\begin{aligned}&\int \left ( x^{2}+3 \right )^{20}2x\: dx=\int u^{20}\: du\\ &=\frac{1}{21}u^{21}+C=\frac{1}{21}\left ( x^{2}+3 \right )^{21}+C \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 2.&\textrm{Selesaikan integral berikut}\\ &\displaystyle \int \left ( x^{4}-x^{2} \right )^{5}\left ( 16x^{3}-8x \right )dx\\\\ &\textbf{Jawab}:\\ &\textrm{Misalkan}\\ &\left\{\begin{matrix} u & = & x^{4} & - & x^{2}&,&\textrm{maka}\\ du & = & 4x^{3} & - &2x&dx \\ 4du & = & 16x^{3} & - & 8x&dx \end{matrix}\right.\\ &\textrm{sehingga dengan integral substitusi}\\ &\begin{aligned}&\displaystyle \int u^{5}.4du=\frac{4}{6}u^{6}+C\\ &=\displaystyle \frac{2}{3}\left ( x^{4}-x^{2} \right )^{6}+C \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 3.&\textrm{Selesaikan integral berikut}\\ &\displaystyle \int \frac{x+2}{x^{2}+4x+4}dx\\\\ &\textbf{Jawab}:\\ &\textrm{Misalkan}\\ &\left\{\begin{matrix} u & = & x^{2} & + & 4x&+&4&,&\textrm{maka}\\ du & = & 2x & + &4&dx \\ \frac{1}{2}du & = & x & + & 2&dx \end{matrix}\right.\\ &\textrm{sehingga dengan integral substitusi}\\ &\begin{aligned}&\displaystyle \int \frac{1}{u}.\frac{1}{2}du=\displaystyle \frac{1}{2}\int \frac{1}{u}du\\ &=\displaystyle \frac{1}{2}\ln u+C=\frac{1}{2}\ln \left ( x^{2}+4x+4 \right )+C \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 4.&\textrm{Selesaikan integral berikut}\\ &\displaystyle \int \frac{e^{3y}}{\left ( 1-2e^{3y} \right )^{2}}dy\\\\ &\textbf{Jawab}:\\ &\textrm{Misalkan}\\ &\left\{\begin{matrix} u & = & 1 & - & 2e^{3y}& ,&\textrm{maka}\\ du & =&- & 6e^{3y} &dy &, \\ -\frac{1}{6}du & = & e^{3y} & dy \end{matrix}\right.\\ &\textrm{sehingga dengan integral substitusi}\\&\begin{aligned}&\displaystyle \int -\frac{1}{6}\frac{1}{u^{2}}du=-\frac{1}{6}\int \frac{du}{u^{2}}\\ &=-\displaystyle \frac{1}{6}\left ( -1 \right )\left ( u^{-1} \right )+C\\ &=\displaystyle \frac{1}{6u}+C\\ &=\displaystyle \frac{1}{6}.\frac{1}{1-2e^{3y}}+C \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 5.&\textrm{Selesaikan integral berikut}\\ &\displaystyle \int 12x\left ( x^{2}+3 \right )^{5}\: dx\\\\ &\textbf{Jawab}:\\ &\begin{aligned}\textrm{Misalkan}\\ u&=x^{2}+3\\ du&=2x\qquad dx \end{aligned}\\ &\textrm{sehingga dengan integral substitusi}\\ &\begin{aligned} &\displaystyle \int 12x\left ( x^{2}+3 \right )^{5}\: dx\\ &=\int \left ( x^{2}+3 \right )^{5}.6.2x\: dx\\ &=6\int \underset{u^{5}}{\underbrace{\left ( x^{2}+3 \right )^{5}}}.\underset{du}{\underbrace{2x\: dx}}\\ &=6\int u^{5}\: du\\ &=6.\displaystyle \frac{u^{5+1}}{5+1}+C\\ &=u^{6}+C\\ &=\left ( x^{2}+3 \right )^{6}+C\end{aligned} \end{array}$.
$\begin{array}{ll}\\ 6.&\textrm{Selesaikan integral berikut}\\ &\displaystyle \int x\sqrt{x-1}\: dx\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\textrm{Misalkan}\qquad u=x-1\\ &du=1\quad dx\: \Leftrightarrow \: du=dx \end{aligned}\\ &\textrm{sehingga dengan integral substitusi}\\ &\begin{aligned} &\displaystyle \int x\sqrt{x-1}\: dx\\&=\int \left ( x-1+1 \right )\sqrt{x-1}\: dx\\ &=\int \left ( \underset{u}{\underbrace{\left (x-1 \right )}}+1 \right )\sqrt{\underset{u}{\underbrace{x-1}}}\: dx \\ &=\int \left ( u+1 \right )\sqrt{u}\: du\\ &=\int \left (u\sqrt{u}+\sqrt{u}\: \right )\: du\\ &=\int \left (u^{\frac{3}{2}}+u^{\frac{1}{2}} \right )\: du\\ &=\displaystyle \frac{1}{\left ( \frac{3}{2}+1 \right )}u^{\left (\frac{3}{2}+1 \right )}+\displaystyle \frac{1}{\left (\frac{1}{2}+1 \right )}u^{\left (\frac{1}{2}+1 \right )}+C\\ &=\displaystyle \frac{2}{5}\left ( x-1 \right )^{\frac{5}{2}}+\displaystyle \frac{2}{3}\left ( x-1 \right )^{\frac{3}{2}}+C\end{aligned} \end{array}$.
$\begin{array}{ll}\\ 7.&\textrm{Selesaikan integral berikut}\\ &\displaystyle \int \displaystyle \frac{x^{5}}{x^{6}+a}\: dx\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\textrm{Misalkan}\\ &u=x^{6}+a\\ &\Leftrightarrow du=6x^{5}\quad dx\\ &\Leftrightarrow \displaystyle \frac{1}{6}du=x^{5}\quad dx \end{aligned}\\ &\textrm{sehingga dengan integral substitusi}\\ &\begin{aligned} &\int \displaystyle \frac{x^{5}}{x^{6}+a}\: dx\\ &=\int \displaystyle \frac{1}{x^{6}+a}.x^{5}\: dx\\ &=\int \displaystyle \frac{1}{\underset{u}{\underbrace{x^{6}+a}}}.\underset{\frac{1}{6}du}{\underbrace{x^{5}\: dx}}\\ &=\displaystyle \frac{1}{6}\int \frac{1}{u}\: du\\ &=\displaystyle \frac{1}{6}\ln \left | u \right |+C\\ &=\displaystyle \frac{1}{6}\ln \left | x^{6}+a \right |+C\end{aligned} \end{array}$.
$\begin{array}{ll}\\ 8.&\textrm{Selesaikan integral berikut}\\ &\displaystyle \int \displaystyle \left ( x^{2}+1 \right )\left ( x^{3}+3x+2 \right )^{5}\: dx\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\textrm{Misalkan}\\ &u=x^{3}+3x+2\\ &\Leftrightarrow du=\left (3x^{2}+3 \right ) dx\\ &\Leftrightarrow du=3\left ( x^{2}+1 \right )dx\\ &\Leftrightarrow \displaystyle \frac{1}{3}du=\left ( x^{2}+1 \right )\: dx \end{aligned}\\ &\textrm{sehingga dengan integral substitusi}\\ &\begin{aligned} \int &\left ( x^{2}+1 \right )\left ( x^{3}+3x+2 \right )^{5}\: dx\\&=\int \left ( x^{3}+3x+2 \right )^{5}.\left ( x^{2}+1 \right )\: dx\\ &=\int \underset{u^{5}}{\underbrace{\left ( x^{3}+3x+2 \right )^{5}}}.\underset{\frac{1}{3}du}{\underbrace{\left ( x^{2}+1 \right )\: dx}}\\ &=\displaystyle \frac{1}{3}\int u^{5}\: du\\ &=\displaystyle \frac{1}{3}.\displaystyle \frac{u^{5+1}}{5+1}+C\\ &=\displaystyle \frac{1}{18}.u^{6}+C\\ &=\displaystyle \frac{1}{18}\left ( x^{3}+3x+2 \right )^{6}+C\end{aligned} \end{array}$.
$\LARGE\colorbox{yellow}{LATIHAN SOAL}$.
$\begin{aligned}&\textrm{Selesaikan soal berikut ini}\\ &\begin{array}{llllll}\\ &\textrm{a}.&\displaystyle \int \displaystyle \frac{4x^{6}+3x^{5}-8}{x^{5}}\: \: dx&\textrm{f}.&\displaystyle \int \displaystyle \frac{\left ( \sqrt{x}+4 \right )^{3}}{\sqrt{x}}\: \: dx\\ &\textrm{b}.&\displaystyle \int \displaystyle \frac{x^{2}}{\sqrt{4-x^{3}}}\: \: dx&\textrm{g}.&\displaystyle \int \displaystyle \frac{x+3}{\sqrt[5]{\left ( x^{2}+6x-1 \right )^{2}}}\: dx\\ &\textrm{c}.&\displaystyle \int x^{3}\left ( x^{4}+10 \right )^{.^{-\frac{2}{3}}}\: dx&\textrm{h}.&\displaystyle \int x^{5}\sqrt{x^{6}+1}\: \: dx \end{array} \end{aligned}$.
DAFTAR PUSTAKA
- Kuntarti, Sulistiyono dan Kurnianingsih, S. 2007. Matematika SMA dan MA untuk Kelas XII Semester 1 Probram IPA Standar ISI 2006. Jakarta: ESIS
- Sharma, S.N., dkk. 2017. Jelajah Matematika SMA Kelas XI Program Wajib. Jakarta: YUDHISTIRA.
- Tung, K.Y. 2012. Pintar Matematika SMA Kelas XII IPA untuk Olimpiade dan Pengayaan Pelajaran. Yogyakarta: ANDI.
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