Jim Coroneos' 100 Integrals ~ 032 ~ ∫1/(1 + cos²x).dx

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Partly to honour Jim, and partly to fulfil an international need, I have decided to produce 100 videos, showing how to solve his 100 integration 'problems.' I hope you find the videos useful!

This thirty-second problem is to evaluate ∫1/(1 + cos²x).dx

You will notice that we have a sum of squares in the denominator, and this suggests an inverse tangent function but, with the cos²x term, there is obviously more "going on" here. Since the reciprocal of cos²x is sec²x and we can produce sec²x in numerator and denominator by dividing by cos²x, we can see a way forward.

This method uses a wonderful property of the tangent function, that it is connected/associated with sec²x via its derivative (d/dx(tanx) = sec²x), and an identity (1 + tan²x = sec²x).

Watch this video to see how it all plays out in this fascinating integral.

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I really appreciated when you explained the thinking behimd the trigonometric substition. i have studied this for a good times and it was the first time it clicked with me. very simple and elegant touch, thanks as always Graeme!

seanki

Very clever way of finding the solution, that dividing numerator and denominator by cos squared of x. We were trying to come close to tan x, indirectly through sec x. That led us to substitution that produced tan as a solution. No escaping from tan in this integral, huh? Yet another multilayered workout and solution. Thank you for explaining it to us, dear Graeme.

MrVoayer

Any chance you are going to do the rest of these?

wjrasmussen

Sir, a doubt
for ∫1/(1 + cos²x).dx
would arctan(cosx) be wrong? if so why?

harishd

i apologise, my previous comment I realised referred to #27

seanki

Jim Coroneos' 100 Integrals ~ 032 ~ ∫1/(1 + cos²x).dx

Jim Coroneos' 100 Integrals ~ 030 ~ ∫1/(3 + 5cosx).dx

Jim Coroneos' 100 Integrals ~ 015 ~ ∫(1+x)/√(1-x-x²).dx

Jim Coroneos' 100 Integrals ~ 021 ~ ∫(cos‾¹x)/√(1-x²).dx

1-10 OF JIM CORONEOS' 100 INTEGRALS SOLVED!

Jim Coroneos' 100 Integrals ~ 016 ~ ∫1/[x²√(1-x²)].dx

Coroneos' 100 Integrals: 050 ∫(x^3)(4+x^2)^(-1/2).dx

Jim Coroneos' 100 Integrals ~ 025 ~ ∫1/[x²(1-x)].dx

Jim Coroneos' 100 Integrals ~ 027 ~ ∫1/(1+x²)².dx

Coroneos' 100 Integrals: 052 ∫(x^2)/(1-x^4).dx

Jim Coroneos' 100 Integrals ~ 005 ~ ∫sinx.sec³x.dx (Three Methods)

Jim Coroneos' 100 Integrals ~ 020 ~ ∫x/(√x+1).dx

Jim Coroneos' 100 Integrals ~ 032 ~ ∫1/(1 + cos²x).dx

Jim Coroneos' 100 Integrals ~ 028 ~ ∫tan³x.dx

Jim Coroneos' 100 Integrals ~ 009 ~ ∫tan‾¹2x.dx

Jim Coroneos' 100 Integrals ~ 023 ~ ∫1/[x(lnx)³].dx

Jim Coroneos' 100 Integrals ~ 026 ~ ∫1/[x²(1+x²)].dx

Jim Coroneos' 100 Integrals ~ 001 ~ ∫x/(x²+4).dx

Jim Coroneos' 100 Integrals ~ 022 ~ ∫√[(x+1)/(x-1)].dx

Jim Coroneos' 100 Integrals ~ 004 ~ ∫sinx.cos³x.dx

Jim Coroneos' 100 Integrals ~ 005 ~ ∫sinx.sec³x.dx

Coroneos' 100 Integrals: 002 ∫x/√(x²+4).dx

Jim Coroneos' 100 Integrals ~ 019 ~ ∫1/[x√(x²-a²)].dx

Jim Coroneos' 100 Integrals ~ 017 ~ ∫1/[x√(a²+x²)].dx

Jim Coroneos' 100 Integrals ~ 008 ~ ∫x.sec²2x.dx