What is the integral of 1/(e^x(5-e^{-x))^3} ?

The integral of 1/(e^x(5-e^{-x))^3} is -1/(2(5-e^{-x))^2}+C

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integral of 1/(e^x(5-e^{-x))^3}

\int \frac{1}{e ^{x} ( 5 - e ^{- x} ) ^{3}} d x

- \frac{1}{2 ( 5 - e ^{- x} ) ^{2}} + C

Solution steps

\int \frac{1}{e ^{x} ( 5 - e ^{- x} ) ^{3}} d x

Apply u-substitution

: \int - \frac{1}{( 5 - u ) ^{3}} d u

= \int - \frac{1}{( 5 - u ) ^{3}} d u

Take the constant out:

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

= - \int \frac{1}{( 5 - u ) ^{3}} d u

Apply u-substitution

: \int - \frac{1}{v ^{3}} d v

= - \int - \frac{1}{v ^{3}} d v

Take the constant out:

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

= - (- \int \frac{1}{v ^{3}} d v)

Apply the Power Rule

: - \frac{1}{2 v ^{2}}

= - (- (- \frac{1}{2 v ^{2}}))

Substitute back

= - (- (- \frac{1}{2 ( 5 - e ^{- x} ) ^{2}}))

Simplify

= - \frac{1}{2 ( 5 - e ^{- x} ) ^{2}}

Add a constant to the solution

= - \frac{1}{2 ( 5 - e ^{- x} ) ^{2}} + C

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What is the integral of 1/(e^x(5-e^{-x))^3} ?
The integral of 1/(e^x(5-e^{-x))^3} is -1/(2(5-e^{-x))^2}+C