What is the integral from 0 to ln(pi) of 2cos(e^x) ?

The integral from 0 to ln(pi) of 2cos(e^x) is 2(\Ci(pi)-0.9460830704)

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integral from 0 to ln(pi) of 2cos(e^x)

\int_{0}^{l n (π)} 2 cos (e^{x}) d x

2 (Ci (π) - 0.9460830704)

Solution steps

\int_{0}^{l n (π)} 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_{0}^{l n (π)} cos (e^{x}) d x

Apply u-substitution

= 2 \cdot \int_{1}^{π} \frac{cos ( u )}{u} d u

Use the nonelementary integral:

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

= 2 [Ci (u)]_{1}^{π}

Compute the boundaries:

Ci (π) - 0.9460830704

= 2 (Ci (π) - 0.9460830704)

What is the integral from 0 to ln(pi) of 2cos(e^x) ?
The integral from 0 to ln(pi) of 2cos(e^x) is 2(\Ci(pi)-0.9460830704)