Find out the answer to "Solve the integral. ∫sec2xetan⁡xdx1=etan⁡x"

lwfrgin

Answered question

2021-10-25

Solve the integral.

\int \frac{\sec^{2} x e^{\tan x} d x}{\sqrt{1 = e^{\tan x}}}

Brittany Patton

Skilled2021-10-26Added 100 answers

Step 1
Given the integral

\int \frac{\sec^{2} x e^{\tan x} d x}{\sqrt{1 = e^{\tan x}}}

We need to find the integral.
Step 2

I = \int \frac{\sec^{2} x e^{\tan x} d x}{\sqrt{1 = e^{\tan x}}}

let

t = 1 + e^{\tan x}

differentiate with respect to x.

\frac{d t}{d x} = e^{\tan x} \sec^{2} x

d t = e^{\tan x} \sec^{2} x d x

I = \int \frac{d t}{\sqrt{t}}

I = \int t^{- \frac{1}{2}} d t

Step 3:integrate the above function.
integrating using power rule

\int x^{n} d x = \frac{x^{n + 1}}{n + 1}

where

n = - \frac{1}{2}

I = \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + C

I = \frac{t^{\frac{1}{2}}}{\frac{1}{2}} + C

I = 2 \sqrt{t} + C

where

t = 1 + e^{\tan x}

I = 2 \sqrt{1 + e^{\tan x}} + C

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