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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