June 28, 20XX
Jolting down, touring, and seeing more memorabilia and advertisements on the main side's walls with lots of key sentences and quotes:
"Donkey Kong, Mario, Luigi, Pauline, Donkey Kong Jr., Qubert, Frogger, and Pitfall Harry celebrate an anniversary that took place on Saturday"
"Mario and Luigi, the famous plumbers, team up with their friends to rescue Princess Peach Toadstool from King Bowser again"
"King Bowser Koopa creates his own TV show to spread mischief and mayhem in the Mushroom Kingdom"
"Mario and his friends journey through the Mushroom Kingdom to defeat the Koopa family and save the magical realm"
"Mario and his friends participate in a series of thrilling challenges and games to prove their skills and save the Mushroom Kingdom"
"Mario and Luigi discover another secret entrance to Dinosaur Land, where with Yoshi and try to rescue Peach again"
"Mario and his friends put on an ice-skating show filled with dazzling performances and adventure"
"Mario and Luigi embark on a quest to rescue Princess Peach"
"Mario and his friends as they battle against King Bowser"
"Mario and Kirby team up in an adventure to save Dream Land and the Mushroom Kingdom from a common threat"
"Mario and his firefighting team extinguishing fires and rescuing citizens in the Mushroom Kingdom"
"Mario and Yoshi explore a vast and magical land, encountering new friends and enemies along the way"
"Comical and lighthearted adventures of Mario and his friends"
"Various events of Mario, his new-founded friends, and their adventures"
"A variety of exciting stories with Mario"
"Mario and Luigi team up with Princess Peach to defeat Bowser and rescue the Mushroom Kingdom"
"Sonic the Hedgehog and his friends at a Gamebook Fair"
"Sonic and his sidekick Tails as they constantly outsmart Dr. Robotnik"
"Sonic and his friends as they fight against the expanding tyranny of Dr. Robotnik"
"Sonic and his friends stopping Dr. Robotnik's plan to conquer the world with the power of Chaos Emeralds"
"Sonic, along with his siblings Sonia and Manic, forming a rock band and getting new members"
"Sonic protecting the planet off-world from various threats while also searching for the Chaos Emeralds to save his friends"
"Sonic and his friends face comedic adventures and battles against Dr. Eggman"
"Sonic and his friends meeting new faces"
"Sonic, Tails, Knuckles, Big, Rouge, and Shadow uncover the importance of new gem relics"
"Knuckles the Echidna as he embarks on solo adventures and explores his unique abilities"
"Sonic the Hedgehog in various adventures"
"Sonic and his allies as they battle against Dr. Robotnik and other villains"
"Sonic and his friends in a range of adventures"
"Sonic and his various friends in another continuation of adventures"
"Sonic and his friends in action-packed adventures"
"Knuckles the Echidna, exploring his solo adventures and uncovering his people's secrets"
"Sonic and his friends' adventures beyond in the stars and places that surrounds this world"
"Sonic the Hedgehog exploring more expanding adventures"
"Sonic and his allies as they battle against the forces of evil, unravel mysteries, and protect continent of Mobius from impending threats"
"Humorous and action-packed adventures with Sonic and his friends"
"Sonic and his friends in various adventures"
"Sonic and his friends expanding their forces' efforts to stop new Dr. Eggman's invasive fronts"
"Sonic and his friends in a new adventure"
"Sonic and his friends as they strive to rebuild and restore their homes after a devastating event, facing new challenges and rediscovering hope along the way"
Math Notes: (filled with academic musings)
Integral pi/2,0 (7 sin x - 2 cos x) dx
(-7 cos x - 2 sin x) | pi/2,0
(-7 cos (pi/2) - 2 sin (pi/2)) - (-7 cos (0) - sin (0))
= -2 - (-7)
= 5
If a polynomial; use the power rule: #(n^x+1 / x+1)
Integral 1,0 3(4x+x^4)(10x^2 + x^5 -2)^6 dx
u=10x^2 + x^5 - 2
du/dx = 20x + 5x^4
du/dx = 5(4x + x^4)
dx = du / 5(4x+x^4)
Integral 1,0 3(4x+x^4) u (du / 5(4x+x^4))
3/5 Integral udu = 3/5 (u^2 / 2) |1,0
= 3/10 (10x^2 + x^5 - 2) |1,0
If a polynomial; use the power rule: #(n^x+1 / x+1)
Integral 1,0 3(4x+x^4)(10x^2 + x^5 - 2)^6 dx
Integral 1,0 3(4x+x^4)(u)^6
dx (du/dx) = 20x + 5x^4 dx
du/(20+5x^4) = (20x+5x^4) dx / 20x + 5x^4
dx = du / 20x+5x^4
Integral 1,0 3(4x+x^4) u^6 du/20x+5x^4
Multiply 1/5 and 5
1/5 Integral 1,0 (5)(3)(4x-x^4)u^6 du/20x+5x^4
3/5 Integral 1,0 (20x-5x^4)u^6 / 20x-5x^4 du
3/5 Integral 1,0 u^6 du
3/5 Integral 1,0 (u^7 / 7) du
Using Power Rule
3/5 (u^7 / 7)
=3/35 (10x^2 + x^5 - 2)^7 |1,0
=3/35 (10(1)^2 + (1)^5 - 2)^7
=409,979.7478
Integral pi/4, 0 8cos(2t) / square root 9-5sin(2t)
Find the U-Sub
u=9-5sin(2t)
du/dt
Integral 1/u^1/2
Integral u^-1/2 du
u= 9-5sin(2t)
du= 9-5sin(2t)
du/dt= -5cos(2t)
Integral pi/4, 0 8cos(2t) / (square root u) dt
Chain Rule
dt (du/dt) = -5cos(2t)(2) dt
dt=du/-10cos(2t)
du/-10cos(2t) = -10cos(2t) / -10cos(2t) dt
Integral pi/4, 0 8cos(2t)/square root u (du/-10cos(2t))
8 Integral pi/4, 0 -10cos(2t)/square root u (du/-10cos(2t))
(-1/10)(8/1) Integral pi/4, 0 -10cos(2t)/square root u (du/-10cos(2t))
-8/10 Integral pi/4, 0 du/square root u
-8/10 Integral pi/4, 0 u^-1/2 du
-8/10 (u^-1/2+1 / -1/2+1) du |pi/4, 0
-8/10 (u^1/2)/(1/2) du |pi/4, u
-8/5 (9-5sin(2t))^1/2 |pi/4, 0
-8/5 (9-5sin(pi/4))^1/2
-8/5 (9-5sin(0))^1/2
=1.059...
Integral 4,1 square root w(e)^1-square root w^3 dw
u=1-square root 1-square root w^3
du=1-square root w^3
dw(du/dw) = -3/2(w^1/2) dw
du/-w = -3/2(w^1/2) dw / -3/2w^1/2
dw = du / -3/2 w^1/2
Integral 4,1 square root w(e)^u du/(-3/2 (w^1/2))
Integral 4,1 square root w(e)^u du(-3/2 (square root w))
-2/3 Integral 4,1 e^u du/(-3/2)
-2/3 Integral 4,1 e^u du
-2/3 (e^u) |4,1
-2/3 e^1-square root w^3 |4,1
(-2/3 e^1-square root 4^3) - (-2/3 e^1-square root 1^3)
=0.66...