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Integral Tak Tentu ~ Matematika SMA | Catatan Nurkholis

Integral Tak Tentu ~ Matematika SMA

Pada materi Turunan/diferensial, kita telah mengetahui bahwa:
$$
begin{aligned}
f(x)=x^2+5xquad&Rightarrowquad f'(x)=2x+5\
f(x)=x^2+5x+2quad&Rightarrowquad f'(x)=2x+5\
f(x)=x^2+5x+9quad&Rightarrowquad f'(x)=2x+5\
f(x)=x^2+5x+Cquad&Rightarrowquad f'(x)=2x+5
end{aligned}
$$
Berapapun nilai $C$ yang kita berikan pada suatu fungsi $f(x)$ akan tetap menghasilkan turunan/diferensial yang sama. Sehingga anti-turunan dari fungsi $f'(x)=2x+5$ dapat dinyatakan dalam bentuk:
$$f(x)=int f'(x)dx=int 2x+5dx = x^2+5x+C$$

Notasi

Secara umum dapat kita tuliskan
$$int x^n dx = frac{x^{n+1}}{n+1}+C,quad nneq-1$$

Sifat

Sifat-sifat Integral tak tentu:

  1. $int [ f(x)pm g(x) ] dx=int f(x) dx pm int g(x) dx$
  2. $int ax^n dx = aint x^n dx = dfrac{a}{n+1}x^{n+1}+C$
  3. $int dx = x + C$
  4. $int a f(x) dx = aint f(x) dx$

Contoh Soal

Tentukan integral tak tentu dari

  1. $3x^2+6x+8$
  2. $5x^4+7x-9$
  3. $x^{frac{1}{2}}+3x^{-frac{1}{2}}+5$

Solusi

  1. $displaystyleint 3x^2+6x+8 dx=x^3+3x^2+8x+C$
  2. $displaystyleint 5x^4+7x-9 dx=x^5+frac{7}{2}x^2-9x+C$
  3. $displaystyleint x^{frac{1}{2}}+3x^{-frac{1}{2}}+5 dx=frac{2}{3}x^{frac{3}{2}}+6x^{frac{1}{2}}+5x+C$
Nah, sekarang mari berlatih!

Latihan Soal

Tentukan integral tak tentu pada bentuk-bentuk berikut!
  1. $displasystyleint 4x+8$
  2. $displasystyleint 3x^2-6x+8$
  3. $displasystyleint 2x^2+3x-4$
  4. $displasystyleint 6x^5-dfrac{5}{2}x^2+13$
  5. $displasystyleint frac{2}{3}x^{frac{1}{2}}+5x^{-frac{1}{2}}-16$
Selamat Belajar!

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