Integral of the Day 7.21.24 | Special DOUBLE Edition! | Math with Professor V

Special DOUBLE edition of your favorite series, "Integral of the Day"! My apologies for double-uploading yesterday, I do pre-film videos for the week and I lost track of the fact I had already published that problem. Nevertheless, I hope you enjoy today's selections, and as always, let me know how you solved these in the comments below. :)

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Professor V

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[siyabongashoba]

wow thank you so much, Prof.V! My last one was long but yours is short and clean! can't wait for more

[dan-florinchereches]

Second problem worked out nice. It looked quite disgusting but worked out elegantly with trig substitution. Never worked out 1/sin X before so I used the second Euler substitution for the radical. The Euler substitutions are gold

[dan-florinchereches]

For the first problem my initial thought was to see what happens if you differentiate the double radical. Turns out to be 1/2* double radical ^(-1/2)*1/2*sqrt(X)^(-1/2) so basically a quarter of the original expression. The integral then is 4*double radical expression +C of we choose nested radical as substitution

[siyabongashoba]

Dear Prof.V I hope you are doing well, for this one, I wrote sqrt(x)= x/sqrt(x) and then let u = 1 + sqrt(x) and du=1/2 * 1/sqrt(x) dx then it gave me 4/7*(sqrt(1 +sqrt(x) )^7 - 8/5 *( sqrt(1 + sqrt(x) ) )^5 + 4/3 *(sqrt( 1 + sqrt(x) ) )^3 + C let me enjoy the video

[siyabongashoba]

Dear Prof.V the second one I wrote 1/x = x^3/x^4 and let u= 1 + x^4 and wrote it in terms of u it was integral of -1/4 *(1/(1+u) )*(1/sqrt(u) ) for that one I let p = sqrt(u) and dp = 1/2 *1/sqrt(u) and now in terms of p it is integral of -1/2 *1/(1-p^2) at this point I apply trig sub by letting p = sin(t) and the end result was -1/2*ln(sec(t) + tan(t) ) + C and then replace the values back to x and I got 1/2* ln( 1/x^2 + sqrt(1 - x^4)/x^2 ) + C what do you think?

[lvxingzhe]

For the second problem I was so close, when I did the triangle at the end I did x^2 = sec theta instead of sin for some reason (not crying). Jk thanks for keeping my calc skills sharp 😄