Teknik Integral : Integral Substitusi Trigonometri


Teknik pengintegralan ini khusus menangani integral-integral yang memuat salah satu bentuk :

\sqrt{a^2-x^2} , \sqrt{a^2 + x^2} , atau \sqrt{x^2-a^2}.

Substitusi yang dilakukan bergantung kepada bentuk-bentuk tersebut dan dirangkum di dalam tabel berikut :

Bentuk Integral

Substitusi

Identitas yang Dipakai

\sqrt{a^2-x^2}

x = a sin \theta

1 – sin2 \theta = cos2 \theta

\sqrt{a^2 + x^2}

x = a tan \theta

 1 + tan2 \theta = sec2 \theta

\sqrt{x^2-a^2}

x = a sec \theta

sec2 \theta – 1 = tan2 \theta

Contoh :

  1. \int \dfrac{\sqrt{x^2-25}}{x} dx = …

    Substitusi x = 5 sec \theta \Leftrightarrow \quad \dfrac{dx}{d\theta} = 5 sec \theta tan \theta , diperoleh :

    \int \dfrac{\sqrt{x^2-25}}{x} dx = \int \dfrac{\sqrt{25sec^2\theta-25}}{5 sec \theta} 5 sec \theta tan \theta d\theta

    = 5 \int \sqrt{sec^2 \quad \theta-1} tan \theta d\theta

    \int 5 tan2 \theta d\theta

    = \int 5(sec2 \theta – 1) d\theta

    = 5 tan \theta – 5\theta + C

    = 5 \sqrt{sec^2 \theta-1} – 5 sec-1 (\dfrac{x}{5}) + C

    = \sqrt{25sec^2 \theta-25} – 5 sec-1 (\dfrac{x}{5}) + C

    = \sqrt{x^2-25} – 5 sec-1 (\dfrac{x}{5}) + C [substitusi x = 5 sec \theta]


Rumus \quad Dasar \quad Integral (tambahan)

8. \int sec2 x dx = tan x + C (BUKTI)

9. \int csc2 x dx = -cot x + C (BUKTI)

10. \int csc x cot x dx = -csc x + C (BUKTI)

\int \dfrac{1}{\sqrt{5-x^2}} dx = …

Substitusi x = \sqrt{5} sin \theta \quad \Leftrightarrow \quad \dfrac{dx}{d\theta} = \sqrt{5} cos \theta , diperoleh :

\int \dfrac{1}{\sqrt{5-x^2}} dx = \int \dfrac{1}{\sqrt{5-5sin^2\theta}} \sqrt{5} cos\theta d\theta

= \int \dfrac{1}{\sqrt{5} \sqrt{1-sin^2\theta}} \sqrt{5} cos\theta d\theta

= \int \dfrac{1}{\sqrt{5} \sqrt{cos^2\theta}} \sqrt{5} cos\theta d\theta

= \int \dfrac{1}{\sqrt{5} \quad cos \theta} \sqrt{5} cos\theta d\theta

= \int d \theta

= \theta + C

= sin-1 (\dfrac{x}{\sqrt{5}}) + C

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