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

The integral of ((e^{-x}+5)^3)/(e^x) is -((e^{-x}+5)^4)/4+C

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

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

- \frac{( e ^{- x} + 5 ) ^{4}}{4} + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int v^{3} d v

= - \int v^{3} d v

Apply the Power Rule

: \frac{v ^{4}}{4}

= - \frac{v ^{4}}{4}

Substitute back

= - \frac{( e ^{- x} + 5 ) ^{4}}{4}

Add a constant to the solution

= - \frac{( e ^{- x} + 5 ) ^{4}}{4} + C

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\int \frac{1}{( x ^{2} - 2 ) ^{\frac{3}{2}}} d x

What is the integral of ((e^{-x}+5)^3)/(e^x) ?
The integral of ((e^{-x}+5)^3)/(e^x) is -((e^{-x}+5)^4)/4+C