What is the integral of (cos(x-1))/(sin^2(x-1)) ?

The integral of (cos(x-1))/(sin^2(x-1)) is -csc(x-1)+C

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integral of (cos(x-1))/(sin^2(x-1))

\int \frac{cos ( x - 1 )}{sin ^{2} ( x - 1 )} d x

- csc (x - 1) + C

Solution steps

\int \frac{cos ( x - 1 )}{sin ^{2} ( x - 1 )} d x

Apply u-substitution

= \int \frac{cot ( u )}{sin ( u )} d u

Rewrite using trig identities

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

Apply u-substitution

= \int \frac{1}{v ^{2}} d v

Apply the Power Rule

= - \frac{1}{v}

Substitute back

= - \frac{1}{sin ( x - 1 )}

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

= - csc (x - 1)

Add a constant to the solution

= - csc (x - 1) + C

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What is the integral of (cos(x-1))/(sin^2(x-1)) ?
The integral of (cos(x-1))/(sin^2(x-1)) is -csc(x-1)+C