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

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

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

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

2 sinh (x) + C

Solution steps

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

Expand

e^{- x} (1 + e^{2 x}) : e^{- x} + e^{x}

= \int e^{- x} + e^{x} d x

Rewrite using trig identities

= \int 2 cosh (x) d x

Take the constant out:

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

= 2 \cdot \int cosh (x) d x

Use the common integral:

\int cosh (x) d x = sinh (x)

= 2 sinh (x)

Add a constant to the solution

= 2 sinh (x) + C

\frac{d}{d x} (\frac{1}{4 x + 4})

f (y) = e^{y}

\int \frac{x - 1}{x ^{2} + x + 1} d x

f (x) = (x (x^{5} + 9)^{3})

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

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