What is the integral of ((e^x))/((2-e^x)^2) ?

The integral of ((e^x))/((2-e^x)^2) is 1/(2-e^x)+C

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integral of ((e^x))/((2-e^x)^2)

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

\frac{1}{2 - e ^{x}} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

= - (- \frac{1}{u})

Substitute back

u = 2 - e^{x}

= - (- \frac{1}{2 - e ^{x}})

Simplify

= \frac{1}{2 - e ^{x}}

Add a constant to the solution

= \frac{1}{2 - e ^{x}} + C

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