What is the integral of 1/(e^xsin^2(e^{-x))} ?

The integral of 1/(e^xsin^2(e^{-x))} is cot(1/(e^x))+C

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integral of 1/(e^xsin^2(e^{-x))}

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

cot (\frac{1}{e ^{x}}) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u ^{2} sin ^{2} ( \frac{1}{u} )} d u

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

Apply u-substitution

: \int - csc^{2} (v) d v

= \int - csc^{2} (v) d v

Take the constant out:

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

= - \int csc^{2} (v) d v

Use the common integral:

\int csc^{2} (v) d v = - cot (v)

= - (- cot (v))

Substitute back

= - (- cot (\frac{1}{e ^{x}}))

Simplify

= cot (\frac{1}{e ^{x}})

Add a constant to the solution

= cot (\frac{1}{e ^{x}}) + C

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What is the integral of 1/(e^xsin^2(e^{-x))} ?
The integral of 1/(e^xsin^2(e^{-x))} is cot(1/(e^x))+C