Evaluate the following integrals.

$\int {e}^{x}\mathrm{tan}\left({e}^{x}\right)dx$

Lorraine Harvey
2021-12-16
Answered

Evaluate the following integrals.

$\int {e}^{x}\mathrm{tan}\left({e}^{x}\right)dx$

You can still ask an expert for help

kalupunangh

Answered 2021-12-17
Author has **29** answers

Step 1

The given integral is$\int {e}^{x}\mathrm{tan}\left({e}^{x}\right)dx$

Step 2

Let$u={e}^{x}$ .

Then$du={e}^{x}dx$

$\int {e}^{x}\mathrm{tan}\left({e}^{x}\right)dx=\int \mathrm{tan}udu$

$=\int \frac{\mathrm{sin}u}{\mathrm{cos}u}du$

$=\int -\frac{1}{v}dv\text{}\text{}\text{}[\because As\sum ev=\mathrm{cos}u\text{}Then\text{}dv=-\mathrm{sin}udu]$

$=-\mathrm{ln}\left|v\right|$

$=-\mathrm{ln}\left|\mathrm{cos}u\right|$

$=-\mathrm{ln}\left|\mathrm{cos}\left({e}^{x}\right)\right|+C$

The given integral is

Step 2

Let

Then

Cheryl King

Answered 2021-12-18
Author has **36** answers

Let

Apply the substitution

Integrate, apply

Back - substitute

Result:

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