Pembuktian Integral csc x cot x dx = -csc x + C


\int csc x cot x dx = \int \frac{1}{sin \quad x} \quad \frac{1}{tan \quad x} dx

= \int \frac{1}{sin \quad x} \quad \frac{cos \quad x}{sin \quad x} dx

= \int \frac{cos \quad x}{sin^2 \quad x} \quad (\frac{d(sin \quad x)}{cos \quad x})

= \int \frac{1}{sin^2 \quad x} d(sin x)

misal : sin x = u kemudian substitusi

= \int \frac{1}{u^2} du

= \frac{1}{-1} \quad \frac{1}{u} + C

kembalikan dalam bentuk trigonometri

= -\frac{1}{sin \quad x} + C

= -csc x + C \blacksquare

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