A. Pengertian
Pengintegralan dari suatu fungsi $f(x)$ berbentuk $\int f(x)dx$ dapat disebut sebagai integral tak tentu dari fungsi $f(x)$ dan jika $F(x)$ adalah anti turunan dari $f(x)$, maka $F(x)dx=F(x)+C$.
$\begin{aligned}&\textrm{Dengan}\\ &\begin{aligned}F(x)&= \textrm{fungsi integral dari}\: \: f(x)\\ f(x)&=\textrm{fungsi yang diintegralkan}\\ C&=\textbf{Konstanta} \end{aligned} \end{aligned}$.
B. Rumus Dasar Integral tak Tentu Fungsi Ajabar
Berikut rumus dasar yang perlu diingat
$\begin{aligned}\bullet \: \: \: &\int dx=x+C\\ \bullet \: \: \: &\int k.\left ( \displaystyle f(x) \right )dx=k\int f(x)dx\\ \bullet \: \: \: &\int \left (f(x)\pm g(x) \right )dx=\int f(x)dx+\int g(x)dx\\ \bullet \: \: \: &\int ax^{n}dx=\displaystyle \frac{a}{n+1}x^{n+1}+C \end{aligned}$.
Sebagai rumus-rumus integral yang lain adalah sebagai berikut
$\begin{array}{ll}\\ \bullet &\displaystyle \int a\: x^{n} dx=\frac{a}{n+1}.x^{n+1}+C, \: \textrm{dengan}\: \: n\neq -1\\ \bullet &\displaystyle \int a\: dx=ax+C\\ \bullet &\displaystyle \int \frac{1}{x}\: dx=\int x^{-1}\: dx=\ln x+C\\ \bullet &\displaystyle \int \left | x \right |\: dx=\frac{1}{2}x\left | x \right |+C\\ \bullet &\displaystyle \int \ln x\: dx=x\ln x-x+C\\ \bullet &\displaystyle \int e^{x}\: dx=e^{x}+C\\ \bullet &\displaystyle \int a^{x}\: dx=\frac{a^{x}}{\ln a}+C\\ \bullet &\displaystyle \int e^{ax}\: dx=\frac{1}{a}.e^{ax}+C\\ \bullet &\displaystyle \int (x^{m}+x^{n}+...+x^{p})\: dx\\ &\qquad =\displaystyle \int x^{m}\: dx+\int x^{n}\: dx+...+\int x^{p}\: d \end{array}$.
$\LARGE\colorbox{yellow}{CONTOH SOAL}$.
$\begin{array}{ll}\\ 1.&\textrm{Selesaikan integral berikut}\\ &\textrm{a}.\quad \displaystyle \int x^{5}\: dx\\ &\textrm{b}.\quad \displaystyle \int 2022x^{5}\: dx\\ &\textrm{c}.\quad \displaystyle \int \displaystyle \frac{1}{x^{2022}}\: dx\\ &\textrm{d}.\quad \displaystyle \int 2022y\: dy\\ &\textrm{e}.\quad \displaystyle \int e^{2022x}\: dx\\ &\textrm{f}.\quad \displaystyle \int 2022^{x}\: dx\\\\ &\textbf{Jawab}:\\ &\textrm{a}.\quad \displaystyle \int x^{5}\: dx=\frac{1}{5+1}.x^{5+1}+C=\frac{1}{6}.x^{6}+C\\ &\textrm{b}.\quad \displaystyle \int 2022x^{5}\: dx=\frac{2022}{5+1}.x^{5+1}+C=337x^{6}+C\\ &\textrm{c}.\quad \displaystyle \int \frac{1}{x^{2022}}\: dx=\int x^{-2022}\: dx\\ &\qquad=\displaystyle \frac{1}{-2022+1}.x^{-2022+1}+C=-\frac{1}{2021}.x^{-2021}+C\\ &\qquad =-\displaystyle \frac{1}{2021x^{2021}}+C\\ &\textrm{d}.\quad \displaystyle \int 2022y\: dy=\frac{2022}{1+1}y^{1+1}+C=1011y^{2}+C\\ &\textrm{e}.\quad \displaystyle \int e^{2022x}\: dx=\frac{1}{2022}.e^{2022x}+C\\ &\textrm{f}.\quad \displaystyle \int 2022^{x}\: dx=\frac{2022^{x}}{\ln 2022}+C \end{array}$.
$\begin{array}{ll}\\ 2.&\textrm{Selesaikan integral berikut}\\ &\textrm{a}.\quad \displaystyle \int \left ( 3x^{2}-x+2-\frac{1}{x}+ \frac{3}{x^{2}}\right )dx\\ &\textrm{b}.\quad \displaystyle \int \left ( x^{2}-2xy+y^{2} \right )dx\\ &\textrm{c}.\quad \displaystyle \int \left | x-1 \right |\: +\left | x-2 \right |\: dx\\\\ &\textbf{Jawab}:\\ &\begin{aligned}\textrm{a}.\quad \displaystyle &\int \left ( 3x^{2}-x+2-\frac{1}{x}+ \frac{3}{x^{2}}\right )dx\\ &=\displaystyle \int 3x^{2}\: dx-\int x\: dx+2\int dx-\int \frac{1}{x}\: dx+3\int \frac{1}{x^{2}}\: dx\\ &=\displaystyle \frac{3}{2+1}x^{2+1}-\frac{1}{1+1}x^{1+1}+2x-\ln x\: +3\left ( \frac{1}{-2+1}x^{-2+1} \right )+C\\ &=\displaystyle \frac{2}{3}x^{3}-\frac{1}{2}x^{2}+2x-\ln x\: -\frac{3}{x}+C \end{aligned}\\&\begin{aligned}\textrm{b}.\quad \displaystyle &\int \left ( x^{2}-2xy+y^{2} \right )dx\\ &=\displaystyle \int x^{2}\: dx-2y\int x\: dx+y^{2}\int dx\\ &=\displaystyle \frac{1}{2+1}x^{2+1}-\frac{2y}{1+1}x^{1+1}+y^{2}.x+C\\ &=\displaystyle \frac{1}{3}x^{3}-x^{2}y+xy^{2}+C \end{aligned}\\ &\begin{aligned}\textrm{c}.\quad \displaystyle &\int \left | x-1 \right |\: +\left | x-2 \right |\: dx\\ &=\displaystyle \frac{(x-1)}{2}\left | x-1 \right |\: +\frac{(x-2)}{2}\left | x-2 \right |+C \end{aligned} \end{array}$.
$\begin{array}{ll}\\ 3.&\textrm{Selesaikan integral berikut}\\ &\textrm{a}.\quad \displaystyle \int 2x^{.^{\frac{2}{3}}}dx\\ &\textrm{b}.\quad \displaystyle \int \frac{1}{3}\sqrt[4]{x^{3}}\: dx\\ &\textrm{c}.\quad \displaystyle \int \displaystyle \frac{x^{4}-4x^{3}}{x^{2}}\: dx\\\\ &\textbf{Jawab}:\\ &\textrm{a}.\quad \displaystyle \int 2x^{.^{\frac{2}{3}}}dx=\displaystyle \frac{2}{\displaystyle \frac{2}{3}+1}x^{.^{\frac{2}{3}+1}}+C=\displaystyle \frac{6}{5}x^{.^{\frac{5}{3}}}+C\\ &\textrm{b}.\quad \displaystyle \int \frac{1}{3}\sqrt[4]{x^{3}}\: dx=\displaystyle \frac{1}{3}x^{.^{\frac{3}{4}}}\: dx=\left (\displaystyle \frac{1}{3} \right ).\displaystyle \frac{1}{\displaystyle \frac{3}{4}+1}x^{.^{\frac{3}{4}+1}}+C\\ &\qquad =\left ( \displaystyle \frac{1}{3} \right )\displaystyle \frac{4}{7}x^{.^{\frac{7}{4}}}+C=\displaystyle \frac{4}{21}x^{.^{\frac{7}{4}}}+C\\ &\textrm{c}.\quad \displaystyle \int \displaystyle \frac{x^{4}-4x^{3}}{x^{2}}\: dx=\int x^{2}-4x\: dx\\ &\qquad =\displaystyle \frac{1}{2+1}x^{2+1}-\displaystyle \frac{4}{1+1}x^{1+1}+C\\ &\qquad =\displaystyle \frac{1}{3}x^{3}-2x^{2}+C \end{array}$.
$\begin{array}{ll}\\ 4.&\textrm{Selesaikan integral berikut}\\ &\textrm{a}.\quad \displaystyle \int \displaystyle \frac{\sqrt{x}(6x^{2}-2x)}{x}dx\\ &\textrm{b}.\quad \displaystyle \int (12x^{2}-4x)(2x+1)\: dx\\\\ &\textbf{Jawab}:\\ &\textrm{a}.\quad \displaystyle \int \displaystyle \frac{\sqrt{x}(6x^{2}-2x)}{x}dx=\displaystyle \int \displaystyle \frac{x^{.^{\frac{1}{2}}}(6x^{2}-2x)}{x}dx\\ &\qquad =\displaystyle \int \displaystyle \frac{6x^{.^{\frac{5}{2}}}-2x^{.^{\frac{3}{2}}}}{x}dx=\displaystyle \int 6x^{.^{\frac{3}{2}}}-2x^{.^{\frac{1}{2}}}dx\\ &\qquad =\displaystyle \frac{6}{\displaystyle \frac{3}{2}+1}x^{.^{\frac{3}{2}+1}}-\frac{2}{\displaystyle \frac{1}{2}+1}x^{.^{\frac{1}{2}+1}}+C\\ &\qquad =\displaystyle \frac{12}{5}x^{.^{\frac{5}{2}}}-\displaystyle \frac{4}{3}x^{.^{\frac{3}{2}}}+C=\displaystyle \frac{12}{5}x^{2}\sqrt{x}-\displaystyle \frac{4}{3}x\sqrt{x}+C\\ &\textrm{b}.\quad \displaystyle \int (12x^{2}-4x)(2x+1)\: dx\\ &\qquad =\displaystyle \int (24x^{3}+4x^{2}-4x)dx\\ &\qquad =\displaystyle \frac{24}{3+1}x^{3+1}+\frac{4}{2+1}x^{2+1}-\frac{4}{1+1}x^{1+1}\: dx\\ &\qquad =6x^{4}+\displaystyle \frac{4}{3}x^{3}-2x^{2}+C \end{array}$.
$\LARGE\colorbox{yellow}{LATIHAN SOAL}$.
$\begin{array}{ll}\\ 1.&\textrm{Selesaikan integral berikut}\\ &\textrm{a}.\quad \displaystyle \int 2022\: dx\\ &\textrm{b}.\quad \displaystyle \int \frac{dx}{2022}\\ &\textrm{c}.\quad \displaystyle \int 2022x\: dx\\ &\textrm{d}.\quad \displaystyle \int -2022x^{2}\: dx\\ &\textrm{e}.\quad \displaystyle \int \left ( x+2022 \right )dx\\ &\textrm{f}.\quad \displaystyle \int \left (-2022x^{3}+2023x^{2}-2024 \right )dx\\ &\textrm{g}.\quad \displaystyle \int x\sqrt{x}\: dx\\ &\textrm{h}.\quad \displaystyle \int \sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{x\sqrt[6]{x}}}}}\: dx\\ &\textrm{i}.\quad \displaystyle \int \frac{2022}{\sqrt[3]{x^{2}}}\: dx\\ &\textrm{j}.\quad \displaystyle \int \frac{2022x}{\sqrt[3]{x^{5}}}\: dx\\ &\textrm{k}.\quad \displaystyle \int \left (2022-2020t+t^{2} \right )dt\\ &\textrm{l}.\quad \displaystyle \int \left (\frac{3}{t^{3}} +\frac{2}{t^{2}} +2022\right )dt\\ &\textrm{m}.\quad \displaystyle \int \left (\sqrt{t} +\frac{1}{2\sqrt{t}} \right )dt\\ &\textrm{n}.\quad \displaystyle \int \left (ay^{4} +by^{2} \right )dy\\ &\textrm{o}.\quad \displaystyle \int \left (4ax^{3}+3bx^{2}+2cx+1 \right )dx\\ &\textrm{p}.\quad \displaystyle \int \frac{x^{2}+2022}{x^{2}}\: dx\\ &\textrm{q}.\quad \displaystyle \int \left (e ^{x}+e^{-x} \right )dx\\ &\textrm{r}.\quad \displaystyle \int e^{2023x}dx\\ &\textrm{s}.\quad \displaystyle \int \frac{dx}{e^{2023x}}dx\\ &\textrm{t}.\quad \displaystyle \int \left ( \sqrt{10^{x}} \right )dx \end{array}$.
$\begin{array}{ll}\\ 2.&\textrm{Selesaikan integral berikut}\\ &\textrm{a}.\quad \displaystyle \int (2022x-2202)dx\\ &\textrm{b}.\quad \displaystyle \int (x^{2}-2x-8) dx\\ &\textrm{c}.\quad \displaystyle \int \sqrt{2x}\: dx\\ &\textrm{d}.\quad \displaystyle \int x^{2}(x+2)(x-1)\: dx \end{array}$.
$\begin{array}{ll}\\ 3.&\textrm{Selesaikan integral berikut}\\ &\textrm{a}.\quad \displaystyle \int 3x\left ( 2x-\displaystyle \frac{1}{x} \right )dx\\ &\textrm{b}.\quad \displaystyle \int \displaystyle \frac{(x^{2}-4)}{x^{2}} dx\\ &\textrm{c}.\quad \displaystyle \int \left ( 2x-\displaystyle \frac{1}{x^{2}} \right )^{2}\: dx\\ &\textrm{d}.\quad \displaystyle \int x^{2}\left ( \displaystyle \frac{(x+4)(x-3)}{x} \right )\: dx \end{array}$.
$\begin{array}{ll}\\ 4.&\textrm{Selesaikan integral berikut}\\ &\textrm{a}.\quad \displaystyle \int \displaystyle \frac{2x^{3}-3x}{\sqrt{x}}dx\\ &\textrm{b}.\quad \displaystyle \int \displaystyle \frac{2}{\sqrt{x}}\left ( 1-\sqrt{x^{2}} \right )^{2} dx\\ &\textrm{c}.\quad \displaystyle \int x^{2}\left ( \sqrt{x}+\displaystyle \frac{1}{\sqrt{x}} \right )^{2}\: dx\\ &\textrm{d}.\quad \displaystyle \int \displaystyle \frac{x^{2}\left ( 2+\sqrt[3]{x^{4}} \right )^{2}}{\sqrt{x}}\: dx \end{array}$.
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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