Get the solution to "Evaluate the following integrals. ∫extan⁡(ex)dx"

Lorraine Harvey

Answered question

2021-12-16

Evaluate the following integrals.

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

kalupunangh

Beginner2021-12-17Added 29 answers

Step 1
The given integral is

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

Step 2
Let

u = e^{x}

.
Then

d u = e^{x} d x

\int e^{x} \tan (e^{x}) d x = \int \tan u d u

= \int \frac{\sin u}{\cos u} d u

= \int - \frac{1}{v} d v [∵ A s \sum e v = \cos u T h e n d v = - \sin u d u]

= - \ln | v |

= - \ln | \cos u |

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

Cheryl King

Beginner2021-12-18Added 36 answers

\int e^{x} {\tan e}^{x} d x

Let

u = e^{x} \Rightarrow d u = - e^{x} \Rightarrow e^{x} d x = - d u

Apply the substitution

\int {\tan e}^{x} e^{x} d x \Rightarrow - \int \tan u d u

Integrate, apply

\int \tan u d u = - \ln | \cos u | + C

- \int \tan u d u = - (- \ln | \cos u |) + C

= \ln | \cos u | + C

Back - substitute

u = e^{x}

= \ln ∣ {\cos e}^{x} ∣ + C

Result:

\ln ∣ {\cos e}^{x} ∣ + C

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