Best solutions to - "Evaluate the following integrals. ∫esin⁡xsec⁡xdx"

nicekikah

Answered question

2021-02-14

Evaluate the following integrals.

\int \frac{e^{\sin x}}{\sec x} d x

Alix Ortiz

Skilled2021-02-15Added 109 answers

The given integral is,

I = \int \frac{e^{\sin x}}{\sec x} d x

Using trigonometric identity, \cos θ = \frac{1}{\sec θ}

I = \int \cos x (e^{\sin x}) d x

Using substitution method,

e^{\sin x} = t

\frac{d}{d x} (e^{\sin x}) = \frac{d t}{d x}

Let

[\frac{d}{d (\sin x)} (e^{\sin x})] [\frac{d}{d x} (\sin x)] = \frac{d t}{d x}

(e^{\sin x}) (\cos x) = \frac{d t}{d x}

(e^{\sin x}) (\cos x) d x = d t

Then the given integral gets transformed as,

I = \int d t = t + C

Putting back the value of t, we get

I = e^{\sin x} + C

Therefore, the value of the given integral is e^{\sin x} + C

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