Evaluate the improper integral \int_{0}^{\infty}\frac{dx}{e^{x}+e^{-x}}

Anonym

Anonym

Answered question

2021-09-29

Evaluate the improper integral 0dxex+ex

Answer & Explanation

Leonard Stokes

Leonard Stokes

Skilled2021-09-30Added 98 answers

Step 1
Consider the integral,
0dxex+ex
Step 2
To solve the indefinite integrals,
dxex+ex=dxex+1ex
=exe2x+1dx
Let, ex=t
exdx=dt
so,
=dtt2+1
=tan1(t)+C
Back  substitution,  ex=t
dxex+ex=tan1(ex)+C
Step 3
To solve the given integral,
Compute the boundaries,
0dxex+ex=[tan1(ex+C)]0
=(tan1(e)+C)(tan1(e0)+C)
=tan1()tan1(1)
=π2π4
=π4

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