From 4f8d02528be1bb372ad942862cdc733b3ee5edad Mon Sep 17 00:00:00 2001 From: "Christoph J. 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wide d u = a'(x) d x \ + A &= integral (2x+7)/(x^2+7x+3) thin d x \ + &= integral (a'(x))/(a(x)) thin d x \ + &= integral (a'(x))/(a(x)) thin d x \ + &= integral 1/(a(x)) dot a'(x) thin d x; wide d u = a'(x) thin d x \ &= integral 1/u d u \ &= ln(u) = ln(x^2+7x+3) checkmark $ + $ - A &= integral (cos(ln(x)))/(x) d x \ - &= integral cos(ln(x)) dot 1/x d x \ - &= integral cos(u) dot u' d x \ + A &= integral (cos(ln(x)))/(x) thin d x \ + &= integral cos(ln(x)) dot 1/x thin d x \ + &= integral cos(u) dot u' thin d x \ &= integral cos(u) dot d u \ &= sin(u) \ &= sin(ln(x)) \ @@ -180,16 +180,23 @@ $ Calculate the following integrals using partial integration: #set enum(numbering: "(a)") - + $ integral (x+2) dot e^(2x) d x $ - + $ integral x^2 dot ln(x) d x $ + + $ integral (x+2) dot e^(2x) thin d x $ + + $ integral x^2 dot ln(x) thin d x $ ]) #set enum(numbering: "(a)") + $ - A &= integral (x+2) dot e^(2x) d x \ - &= integral u(x) dot v'(x) d x \ - &= u(x) dot v(x) - integral u'(x) dot v(x) d x \ - &= (x+2) dot (e^(2x) dot 1/2x) - integral e^(2x) d x \ - &= (x+2) dot (e^(2x) dot 1/2x) - e^(2x) dot 1/2x \ - &= x dot (e^(2x) dot 1/2x) + e^(2x) dot 1/2x \ -$ + u(x) &= x => u'(x) = 1; space v(x) = e^(2x) dot 1/2 x => v'(x) = e^(2x) \ + A &= integral (x+2) dot e^(2x) thin d x \ + &= integral x dot e^(2x) thin d x + integral 2 dot e^(2x) thin d x wide \ + &= integral x dot e^(2x) thin d x + 2 dot integral e^(2x) thin d x \ + &= integral x dot e^(2x) thin d x + 2 dot 1/2 e^(2x) \ + &= integral x dot e^(2x) thin d x + e^(2x) \ + &= integral u(x) dot v'(x) thin d x + e^(2x) \ + &= [u(x) dot v(x)] - integral u'(x) dot v(x) thin d x + e^(2x) \ + &= x dot e^(2x) dot 1/2 - integral 1 dot e^(2x) dot 1/2 thin d x + e^(2x) \ + &= 1/2 x dot e^(2x) - e^(2x) 1/4 + e^(2x) \ + &= e^(2x) dot (1/2 x - 1/4 + 1) \ + &= 1/2 e^(2x) dot (x - 1/2 + 2) \ + &= 1/2 e^(2x) dot (x + 3/2) checkmark +| $