June 30, 20XX
The room was filled with excitement and anticipation as our beloved heroes, Mario, Luigi, Yoshi, and Toad, prepare to embark on a journey to the continent of Mobius. Not only that, but they will be joined by none other than Sonic the Hedgehog and his remarkable team, including Tails, Amy, Knuckles, Cream and Cheese, Rouge, Omega, and Shadow. The prospect of such an alliance is awe-inspiring, and I can't help but feel privileged to be part of this extraordinary adventure.
The air buzzes with a unique energy as our heroic group gathers at the Mushroom Kingdom's borders. Mario, as always, leads with his unwavering determination and optimism, his iconic red hat adorned with the letter 'M' glinting in the sunlight. Luigi stands tall beside him, his green hat matching his brother's, filled with an equal measure of courage and curiosity. Yoshi, with his vibrant and cheerful presence, radiates an infectious joy that uplifts us all. And then there's Toad, ever the reliable companion, his loyalty and resourcefulness making him an essential member of our team.
As we gather, we are greeted by our newfound allies from the continent of Mobius. Sonic, the fleet-footed blue hedgehog, exudes an air of cool confidence, his trademark red shoes gleaming. Tails, his loyal companion and a brilliant inventor, brings with him an undeniable sense of curiosity and knowledge. Amy, the courageous and determined pink hedgehog, stands by Sonic's side, her trusty Piko Piko Hammer at the ready. Knuckles, the powerful guardian of the Master Emerald, brings his incredible strength and unwavering dedication to our cause.
Cream and Cheese, the adorable and kind-hearted rabbit and Chao duo, fill the air with innocence and purity, reminding us of the beauty in the world. Rouge, the enigmatic bat, with her charming wit and expert thieving skills, adds an element of mystery to our group. Omega, the powerful and battle-hardened robot, stands tall and ready, his red eyes glowing with a fierce determination. And last but not least, Shadow, the brooding and complex hedgehog, brings his unparalleled strength and a sense of profound purpose.
Together, our combined forces form an alliance of unprecedented proportions, uniting the common goal in a quest for justice and peace. We stand at the cusp of an extraordinary adventure, ready to explore the uncharted territories of Mobius and face whatever challenges lie ahead.
Meanwhile as we all aboard an air vehicle, I read my academic notes again to eliminate some time.
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...