Expert Answer to "Solve the integral. ∫esec⁡(x)sin⁡(x)cos2(x)dx"

texelaare

Answered question

2021-10-29

Solve the integral.

\int \frac{e^{\sec (x) \sin (x)}}{\cos^{2} (x)} d x

Latisha Oneil

Skilled2021-10-30Added 100 answers

Step 1
We need to evaluate the following integral

I = \int \frac{e^{\sec (x) \sin (x)}}{\cos^{2} (x)} d x

...(1)
Step 2
Let

\sec (x) = t \Rightarrow \frac{d t}{d x} = \frac{d}{d x} (\sec (x)) \Rightarrow \frac{d t}{d x} = \frac{\sin (x)}{\cos^{2} (x)} d x = d t

and substitute these values in equation (1). We get,

I = \int e^{t} d t

\Rightarrow I = e^{t} + C

\Rightarrow I = e^{\sec (x)} + C

Hence, the value of the integral is

\int \frac{e^{\sec (x) \sin (x)}}{\cos^{2} (x)} d x = e^{\sec (x)} + C

Required value

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