July 4, 20XX
Today has been an extraordinary day filled with excitement and newfound hope. Mario and Sonic have made a groundbreaking decision that could potentially change the course of the mission against the forces of evil. They have resolved to expand our alliance and gather more allies to fight alongside us, and we have set our sights on three remarkable places: Corneria, Mega City, and the futuristic Earth's military forces.
Mario and Sonic have always been known for our bravery and resilience, but we acknowledge that the challenges ahead require a united front. We have been battling against the forces of darkness for what feels like an eternity, but the time has come to forge stronger bonds and rally an even larger army of heroes.
First on the list is Corneria, the home planet of Fox McCloud and his team of Star Fox. We believe that their expertise in interstellar combat and aerial superiority will be a valuable addition to our cause. We have heard tales of their heroic exploits, and their determination to protect the galaxy aligns perfectly with our own goals. Mario and Sonic are eager to meet Fox and his team, hoping that together we can strike a significant blow against the forces of evil.
Next, they have their sights set on Mega City, a sprawling metropolis filled with extraordinary beings. Here, they hope to recruit the iconic Mega Man and his friends, whose unmatched skills in battle and ability to harness the powers of defeated foes make him a formidable ally. Mega Man's unwavering dedication to justice and his desire to rid the world of evil resonate deeply with both Mario and me. We are eager to reach out to him and discuss our mission, hoping that he will join our cause.
Lastly, our attention turns towards recent reinforcements. We have caught wind of a military force from the futuristic Earth, armed with advanced technology and formidable weaponry. This force has faced its own battles against ruthless villains and malevolent forces, making them experienced fighters who could greatly aid our cause. Their knowledge and futuristic resources could potentially turn the tide of this ongoing conflict in our favor. It is our hope that by uniting our efforts, we can bring an end to this era of darkness.
Of course, reaching out to these potential allies will not be an easy task. Mario and Sonic are aware that it will require diplomacy, persuasive words, and a deep understanding of the challenges each of these realms faces. We are prepared to embark on this journey, ready to put our best foot forward and forge strong alliances that will lead us to victory.
As I pen down these words, I feel a renewed sense of determination and optimism. We are fully aware of the trials that lie ahead, but with the support of new allies from Corneria, Mega City, and futuristic Earth's military forces, we stand a chance against the encroaching darkness.
May this journal entry mark the beginning of an extraordinary chapter in our ongoing battle against evil. Mario and Sonic are ready to face the challenges that lie ahead, and with the power of friendship and cooperation, we believe that victory is within our grasp.
I read my academic notes again (with Dr. E. Gadd and Tails) 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...