What is the integral of ((1+cot^2(z))cot(z))/(csc(z)) ?

The integral of ((1+cot^2(z))cot(z))/(csc(z)) is -csc(z)+C

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integral of ((1+cot^2(z))cot(z))/(csc(z))

\int \frac{( 1 + cot ^{2} ( z ) ) cot ( z )}{csc ( z )} d z

- csc (z) + C

Solution steps

\int \frac{( 1 + cot ^{2} ( z ) ) cot ( z )}{csc ( z )} d z

Simplify

\frac{( 1 + cot ^{2} ( z ) ) cot ( z )}{csc ( z )} : csc^{2} (z) cot (z) sin (z)

= \int csc^{2} (z) cot (z) sin (z) d z

Rewrite using trig identities

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

Apply u-substitution

: \int \frac{1}{u ^{2}} d u

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

Apply the Power Rule

: - \frac{1}{u}

= - \frac{1}{u}

Substitute back

u = sin (z)

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

Use the basic trigonometric identity:

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

= - csc (z)

Add a constant to the solution

= - csc (z) + C

\int z e^{y} d y

\frac{d y}{d t} = 1 + t + y + t y

\frac{d}{d x} (x^{2} + 2 x - 8)

\frac{\partial}{\partial x} (e^{- x} tan (y))

e^{- 2 t} sin (3 π t)

What is the integral of ((1+cot^2(z))cot(z))/(csc(z)) ?
The integral of ((1+cot^2(z))cot(z))/(csc(z)) is -csc(z)+C