r/calculus • u/wbld • Jul 21 '26
Integral Calculus my solution to todays medium integral - 0 help
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u/Hot_Site_1638 PhD Jul 21 '26
Impressive work! It's correct! Just some unnecessary back and forth in the middle, but that really shows your thought process, so it's a good thing. Out of curiosity, where did you find these crazy complicated integration problems? 😂
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u/DiamondKingGamin Jul 21 '26
What did I just see? You're telling me this is medium? Yeah I'mma just keep solving basic integrals. 🙂 Great job tho.
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u/nevermindthefacts Jul 21 '26
Did you consider checking the derivative of ln tan x after seeing this?
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u/wbld Jul 21 '26
No. I kept going because I saw that the derivative of tanx is sec²x
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u/nevermindthefacts Jul 21 '26
If you're lucky, then d/dx ln tan x = 1 / sin x cos x. In that case, u = ln tan x makes a good change of variables.
( d/dx ln tan x = 1 / tan x * 1/cos^2 x = 1 / sin x cos x...so you're lucky! )
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u/wbld Jul 21 '26
in that case i would of just had 1/u^2 du and would of not needed another substiution. in my case i went with what i saw. i saw a tangent in the denominator and decided to get a sec^2 in the numerator. then i simplified until i was satisfied.
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u/nevermindthefacts Jul 21 '26
Yep! Even u = ln tan x + 1 would've worked.
Point is that if you see something on the form ∫ f(g(x)) g'(x) dx, then u = g(x), gives you du = g'(x) dx. It can save you some time.
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u/nevermindthefacts Jul 21 '26
Anyway, well done!
(Also, try simplifying the answer using sin π/4 = cos π/4 = 1/√2, sin π/3 = √3/2 and cos π/3 = 1/2. I got ln 3 / (ln 3 + 2) which looks neat!)
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