This Monster Has a One-Line Answer

This integral looks impossible: the integral of cosine of two times inverse cotangent of the square root of one minus x over one plus x, dx. A nested inverse cotangent, a square root, a fraction. Yet the answer is one line: minus x squared over two. The trick is to name the inner angle. Let phi be that inverse cotangent, so cotangent squared phi equals one minus x over one plus x, and the integrand is just cosine of two phi. Write cosine of two phi through cotangent: it equals cotangent squared phi minus one, over cotangent squared phi plus one. Substitute, and the algebra collapses: the numerator becomes minus two x over one plus x, the denominator becomes two over one plus x, the one plus x and the two cancel, and the whole monster equals minus x. So the integral is just the integral of minus x dx, which is minus x squared over two plus C. It looked scary, but one identity opened it up. Chapters: 00:00 A monster that takes one line 00:35 Name the inner angle 01:06 cos 2 phi through cotangent 01:51 Substitute cot squared phi 02:20 Simplify the numerator 02:49 The fraction collapses to minus x 03:29 Integrate: minus x squared over two #math #calculus #integration Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com Stream the music on Spotify: https://open.spotify.com/artist/2SRhE... GEMDAS, order of operations explained, simplify expressions, evaluate expressions, parentheses exponents, multiplication and division, addition and subtraction, viral math problem, viral math question, tricky math problem, can you solve this, 90 percent get this wrong, math problem most people get wrong, hardest math problem, simple math problem, math riddle, math quiz, math test, math exam, math question, math problem of the day, brain teaser, brain training, brain exercise, IQ test, homeschool math, elementary math, middle school math, high school math, math for kids, math for adults, GCSE maths, SAT math, math help, speed math, math shortcuts, math hacks, Vedic math, math made easy