From 595773d2fe65f068e209cd68a79758a512a9e45e Mon Sep 17 00:00:00 2001 From: "Christoph J. Scherr" Date: Fri, 6 Sep 2024 15:43:06 +0200 Subject: [PATCH] integrals are easy actually???? --- build/main.pdf | Bin 437950 -> 444550 bytes src/exercise/1.typ | 55 ++++++++++++++++++++++++++++++++++++++++----- 2 files changed, 50 insertions(+), 5 deletions(-) diff --git a/build/main.pdf b/build/main.pdf index 509f813f47f65e0fa4fc46ee2e35f27bb725926f..a26af20d29e25e187a7990c256bdabc026b4e30e 100644 GIT binary patch delta 110798 zcmZs>byQrz(Qkq>y1P~=z9zlFXVR8eUkb#aY z&~QL+Nl*xr9!Zj{#7G2ad1-YmFllLd^gRsRooqa8?P&RhQtZ)aKwU8}UxL<2P&kqv zzdRQN@sK|!ZOnRpiSqBiKISFie^f8ht;D768zs}{vZ1a@`8GPqY(5 zeI+kVE6K|%B$q;mN!0WbmH9cN-2bc*k(I{$kK(`dB69yFDg2jCzFF`&>L3c)zisou z-p0||8&gPx_Jf^+lly;(AS5a(N3+xm)EHD0!GCt!MI(73CMGNjdWS?K14&V$FoWn` zq2{2lI5l^_Ld6{isFu5V#osVEMSOY@hoSPeaH$uK{JT?nxo|Q};ka~(%rzqTSYuk* z`?&K>1eWmopyMyp$-n8oe_<>J;V*veSZsLC!XkE^<8%DbeH?tO3>lp}U~`$PIx}-Q z4ivwKZdlH1mtYJU--T3!)(>(T_&_q26-tTO5V?X(d0u=7lk0Cc6eH9zIeOB0j z*u6jQ`s*A^N)AmA+=#xdI#I?lyfU7s@!g@_N%CKTvFE}Lv<|O_vIH`0`pA0&qXbgV 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((n^4-2)(n^3+1) + (n^3(3-n^2))(n^2+4)) / ((n^2+4)(n^3+1)) \ &= (n^7+n^4-2n^3-2 + 3n^5+12n^3-n^7-4n^5) / ((n^2+4)(n^3+1)) \ &= (-n^5+n^4+10n^3-2) / (n^5+n^2+4n^3+4) \ &= (-1+1/n+10/(n^2)-2/(n^5)) / (1+1/(n^3)+4/(n^2)+4/(n^5)) \ - => lim_(n -> infinity) a_n + => lim_(n -> infinity) a_n &= lim_(n -> infinity) (-1+1/n+10/(n^2)-2/(n^5)) / (1+1/(n^3)+4/(n^2)+4/(n^5)) \ &= lim_(n -> infinity) -1/1 = 1 checkmark $ @@ -72,7 +72,7 @@ $ + $ A &= sum_(n=1)^infinity (2^n n!)/(n^n) \ a_n &= (2^n n!)/(n^n) #text("Ratio Test") \ - => lim_(n -> infinity) a_n + => lim_(n -> infinity) a_n &= lim_(n -> infinity) abs((a_(n+1))/(a_n)) \ &= lim_(n -> infinity) abs( ( (2^(n+1) (n+1)!)/((n+1)^(n+1)) )/( (2^n n!)/(n^n) )) \ &= lim_(n -> infinity) abs( ( 2^(n+1) dot (n+1)! dot n^n )/( 2^n dot n! dot (n+1)^(n+1) )) \ @@ -88,7 +88,7 @@ $ + $ a_n &= 1/(n^n) #text("Root Test") \ - root(n, abs(1/(n^n))) &= 1/n -> 0 < 1 \ + root(n, abs(1/(n^n))) &= 1/n -> 0 < 1 \ &=> a_n #text(" diverges") checkmark $ @@ -109,6 +109,7 @@ $ #set enum(numbering: "(a)") + $ + f(x) &= ln((e^x) / (1+e^x)) \ e(x) = e^x wide a(x) &= (e^x)/(1+e^x) wide b(x) = 1+e^x \ ln: RR^+ |-> RR wide e&: RR |-> RR^+ wide => f: RR |-> RR <=> DD_f = RR \ @@ -124,3 +125,47 @@ $ &= (1+e^x)/([1+e^x]^2) \ &= 1/(1+e^x) checkmark $ ++ $ + D_f &= RR \\ {x = 1}\ + f(x) &= ((x+1)/(x-1))^2 \ + a(x) = x+1 wide b(x) = x-1 quad&quad c(x) = a(x)/b(x) wide d(x) = x^2 \ + a'(x) = 1 wide b'(x) &= 1 wide d'(x) = 2x \ + c'(x) &= (b' dot a - b dot a')/(a^2) \ + &= (x-1 - (x+1))/((x-1)^2) \ + &= (-2)/((x-1)^2) \ + f'(x) &= d'(c(x)) dot c'(x) \ + &= 2 dot (x+1)/(x-1) dot (-2)/((x-1)^2) \ + &= -4 dot (x+1)/((x-1)^3) checkmark +$ + +#pagebreak() +=== Exercise 4 @Exercise[1, 4] +#block( + fill: luma(230), + inset: 8pt, + radius: 4pt, +[ + Calculate the following integrals using substitution: + + #set enum(numbering: "(a)") + + $ integral (2x+7)/(x^2+7x+3) d x $ + + $ integral (cos(ln(x)))/(x) d x $ +]) + +#set enum(numbering: "(a)") ++ $ + A &= integral (2x+7)/(x^2+7x+3) d x \ + &= integral (a'(x))/(a(x)) d x \ + &= integral (a'(x))/(a(x)) d x \ + &= integral 1/(a(x)) dot a'(x) d x; wide d u = a'(x) 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 \ + &= integral cos(u) dot d u \ + &= sin(u) \ + &= sin(ln(x)) \ +$