What is the integral of e^{cot(x)}cot^2(x)csc^2(x) ?

The integral of e^{cot(x)}cot^2(x)csc^2(x) is -e^{cot(x)}cot^2(x)+2e^{cot(x)}cot(x)-2e^{cot(x)}+C

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integral of e^{cot(x)}cot^2(x)csc^2(x)

\int e^{c o t (x)} cot^{2} (x) csc^{2} (x) d x

- e^{c o t (x)} cot^{2} (x) + 2 e^{c o t (x)} cot (x) - 2 e^{c o t (x)} + C

Solution steps

\int e^{c o t (x)} cot^{2} (x) csc^{2} (x) d x

Apply u-substitution

: \int - e^{u} u^{2} d u

= \int - e^{u} u^{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int e^{u} u^{2} d u

Apply Integration By Parts

: u^{2} e^{u} - \int 2 u e^{u} d u

= - (u^{2} e^{u} - \int 2 u e^{u} d u)

\int 2 u e^{u} d u = 2 (e^{u} u - e^{u})

= - (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = cot (x)

= - (cot^{2} (x) e^{c o t (x)} - 2 (e^{c o t (x)} cot (x) - e^{c o t (x)}))

Simplify

- (cot^{2} (x) e^{c o t (x)} - 2 (e^{c o t (x)} cot (x) - e^{c o t (x)})) : - e^{c o t (x)} cot^{2} (x) + 2 e^{c o t (x)} cot (x) - 2 e^{c o t (x)}

= - e^{c o t (x)} cot^{2} (x) + 2 e^{c o t (x)} cot (x) - 2 e^{c o t (x)}

Add a constant to the solution

= - e^{c o t (x)} cot^{2} (x) + 2 e^{c o t (x)} cot (x) - 2 e^{c o t (x)} + C

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What is the integral of e^{cot(x)}cot^2(x)csc^2(x) ?
The integral of e^{cot(x)}cot^2(x)csc^2(x) is -e^{cot(x)}cot^2(x)+2e^{cot(x)}cot(x)-2e^{cot(x)}+C