Question: "Find the indefinite integral. ∫e−xtan⁡(e−x)dx"

Suman Cole

Answered question

2021-11-09

Find the indefinite integral.

\int e^{- x} \tan (e^{- x}) d x

FieniChoonin

Skilled2021-11-10Added 102 answers

Step 1
Evaluate the indefinite integral.

\int e^{- x} \tan (e^{- x}) d x = \int - \tan (u) d u [∵ u = e^{- x} \Rightarrow d u = - e^{- x} d x]

= - \int \frac{\sin (u)}{\cos (u)} d u

= - \int - \frac{1}{v} d v [∵ v = \cos u \Rightarrow d v = - \sin u d u]

= - \ln (v) + C

Step 2
Substitute back

v = \cos u

in the integral as follows.

\int e^{- x} \tan (e^{- x}) d x = - \ln | \cos u | + C

= - \ln | \cos (e^{- x}) | + C [∵ u = e^{- x}]

Step 3:
Answer:
The value of the indefinite integral

\ln | \cos (e^{- x}) | + C

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