Pembuktian Integral csc x dx = ln |csc x – cot x| + C


\int csc x dx = \int csc x \frac{csc \quad x - cot \quad x}{csc \quad x - cot \quad x} dx

.

misal : u = csc x – cot x

u = \frac{1}{sin \quad x} \frac{cos \quad x}{sin \quad x}

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

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

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

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

du = (csc x cot x – csc2 x) dx

.

= \int \frac{(csc^2 \quad x + csc \quad x \quad cot \quad x)}{csc \quad x + cot \quad x} dx

= \int \frac{du}{u}

= ln |u| + C

= ln |csc x – cot x| + C \blacksquare

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