[Verse 1] When functions need their antiderivatives found Power rule starts the mathematical sound X to the n becomes n plus one in the fraction Divide by that new power, that's integration action But when n equals negative one, there's a twist Natural log appears where polynomials missed [Chorus] Integration's an art, not a machine You've got to see the hidden structure in between Power rule and logarithms, exponentials stay the same Sine becomes negative cosine in this game Cosine turns to sine, that's the pattern we retain Arctangent waits for one plus x squared's domain [Verse 2] E to the x integrates to itself with ease One over square root one minus x squared needs arcsine, please When products multiply, try integration by parts U times dv equals uv minus integral starts With v times du, reverse the product rule's art [Chorus] Integration's an art, not a machine You've got to see the hidden structure in between Power rule and logarithms, exponentials stay the same Sine becomes negative cosine in this game Cosine turns to sine, that's the pattern we retain Arctangent waits for one plus x squared's domain [Bridge] Substitution's the chain rule running backwards through time Let u equal g of x, then du equals g prime F of g times g prime becomes f of u so clean Transform the variable, make the complex routine Logarithms bridge the polynomial and exponential scene [Verse 3] No algorithm exists to solve them all Integration demands creativity's call Pattern recognition guides your mathematical sight Choose substitution when derivatives unite Or parts when products need a different light [Outro] Don't forget your constant C at every turn These techniques are heuristics, not rules you simply learn The art of integration takes practice to discern
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