I liked both solutions. I find yours more intuitive, it's the approach I would have used. bprp's solution is "accidentally elegant" in that the solution suddenly simplifies at the end when you transform the infinite sum into e^lnx, but you never see it coming while you're churning through the maths. Your way you feel more in control and methodical all the way through.
Thanks mate. I like my solution developments to feel that way cuz although it's sometimes nice to feel surprised by the math I like my surprises to feel more satisfying. But hey that's integral calculus for ya....its an art rather than a bunch of mechanical steps.
@@maths_505 I compare the rigor in math to the rigor in painting or playing violin, or knowing the deep details of a language's grammar and history being needed to make literature. Mathematics is an art, just a different part of the brain is used in the rigor than other arts with the same DEGREE of rigor.
Elegant & creative solution. Your choice of substitution worked out well. Applying the power of inverse operations. From additive inverse to multiplicative inverse. Exponential function is the inverse of ln function. Integration and differentiation are inverse operations. Square and square root, Laplace Transform and Inverse Laplace Transform, matrix and its inverse, real function and its inverse and the list continues. Possibly complex function and its inverse. The profound relatedness of Mathematical definitions and concepts is amazing. Expanding a condensed ln integral using a substitution of its inverse leads to an elegant solution. I think a holistic and intuitive approach to solving problems is useful. This nested radical of ln resembles a telescoping series. The e^t came in useful to replace the infinite series exponent of the integrand.
A little bit easier if you just take out a 1/e and then reverse the subst from before to do the final integration imo, but this is what I thought of too!
I mean, you are given like 4 minutes to integrate this, so the problem must be doable. And you know, there's not a lot of pure calculus problems in this range, so it usually is just algebra, with the help of calculus.
I liked both solutions. I find yours more intuitive, it's the approach I would have used.
bprp's solution is "accidentally elegant" in that the solution suddenly simplifies at the end when you transform the infinite sum into e^lnx, but you never see it coming while you're churning through the maths. Your way you feel more in control and methodical all the way through.
Thanks mate.
I like my solution developments to feel that way cuz although it's sometimes nice to feel surprised by the math I like my surprises to feel more satisfying. But hey that's integral calculus for ya....its an art rather than a bunch of mechanical steps.
@@maths_505 I compare the rigor in math to the rigor in painting or playing violin, or knowing the deep details of a language's grammar and history being needed to make literature. Mathematics is an art, just a different part of the brain is used in the rigor than other arts with the same DEGREE of rigor.
Elegant & creative solution. Your choice of substitution worked out well. Applying the power of inverse operations. From additive inverse to multiplicative inverse. Exponential function is the inverse of ln function. Integration and differentiation are inverse operations. Square and square root, Laplace Transform and Inverse Laplace Transform, matrix and its inverse, real function and its inverse and the list continues. Possibly complex function and its inverse. The profound relatedness of Mathematical definitions and concepts is amazing. Expanding a condensed ln integral using a substitution of its inverse leads to an elegant solution. I think a holistic and intuitive approach to solving problems is useful. This nested radical of ln resembles a telescoping series. The e^t came in useful to replace the infinite series exponent of the integrand.
Beautiful
Enjoyed your solution! 😊👍
Yet to see bprp’s solution, but this method was extremely satisfying.
I think this is the more intuitive approach, or it's atleast what came to my mind instantly
Pretty nice solution!
very nice
Nice video 😊
A little bit easier if you just take out a 1/e and then reverse the subst from before to do the final integration imo, but this is what I thought of too!
Dear Maths 505,
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Nice bro
Nice sir
I did that in blackpenredpen's way, but this way is not bad.
Yooo just as promised :D
Good🎉
This approach feels as if you are a JEE student
answer forty one
yes
I feel this is an problem in algebra, less related to integrals
I mean, you are given like 4 minutes to integrate this, so the problem must be doable. And you know, there's not a lot of pure calculus problems in this range, so it usually is just algebra, with the help of calculus.
You lost me at "A few days ago..."
Wrong.
Final integral is not easy to solve. And you solved it easily with wrong way
It's just ₀∫² eˣ⁻¹ dx
e=x^{1/(lnx)}
eˣ⁻¹ = x^{(x-1)/lnx}
(x-1)/lnx=(eˡⁿˣ - 1)/lnx