Pembuktian Integral sec x dx = ln |sec x + tan x| + C


\int sec x dx = \int sec x \frac{sec \quad x + tan \quad x}{sec \quad x + tan \quad x} dx

.

misal : u = sec x + tan x

u = \frac{1}{cos \quad x} + \frac{sin \quad x}{cos \quad x} (mencari turunannya menggunakan Turunan Aturan Pembagian)

\frac{du}{dx} = \frac{0(cos \quad x)-1(-sin \quad x)}{cos^2 \quad x} + \frac{cos \quad x(cos \quad x)-sin \quad x(-sin \quad x)}{cos^2 \quad x}

= \frac{0(cos \quad x)-1(-sin \quad x)}{cos^2 \quad x} + \frac{cos \quad x(cos \quad x)-sin \quad x(-sin \quad x)}{cos^2 \quad x}

= \frac{sin \quad x}{cos^2 \quad x} + \frac{cos^2 \quad x+sin^2 \quad x}{cos^2 \quad x}

= \frac{1}{cos \quad x} \frac{sin \quad x}{cos \quad x} + \frac{1}{cos^2 \quad x}

du = (sec x tan x + sec2 x) dx

.

= \int \frac{(sec^2 \quad x + sec \quad x \quad tan \quad x)}{sec \quad x + tan \quad x} dx

substitusi u dan du, sehingga

= \int \frac{du}{u}

= ln |u| + C

kembalikan kedalam bentuk trigonometri lagi

= ln |sec x + tan x| + C \blacksquare

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2 comments on “Pembuktian Integral sec x dx = ln |sec x + tan x| + C

  1. Ping-balik: Problem (23) : Integral | Math IS Beautiful

  2. Ping-balik: Integral dari Invers Fungsi Trigonometri | Math IS Beautiful

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