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

The integral of 2cos(e^x) is 2\Ci(e^x)+C

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integral of 2cos(e^x)

\int 2 cos (e^{x}) d x

2 Ci (e^{x}) + C

Solution steps

\int 2 cos (e^{x}) d x

Take the constant out:

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

= 2 \cdot \int cos (e^{x}) d x

Apply u-substitution

= 2 \cdot \int \frac{cos ( u )}{u} d u

Use the nonelementary integral:

\int \frac{cos ( x )}{x} d x = Ci (x)

= 2 Ci (u)

Substitute back

u = e^{x}

= 2 Ci (e^{x})

Add a constant to the solution

= 2 Ci (e^{x}) + C