Calculus study realm

Sure, go.

7) ∫(0 to ∞) dx/(x^2 + 4) = lim(c-->∞) ∫(0 to c) dx/(x^2 + 4) = lim(c-->π/4) ∫(0 to c) 2(sec@)^2 /(4(tan@)^2 + 4) d@ = lim(c-->π/2) ∫(0 to c) 1/2 d@ = lim(c-->π/2) @/2|(0 to c) = lim(c-->π/2) c/2 = π/4

let x = 2tan@
dx = 2(sec@)^2 d@

8) ∫(3 to ∞) dx/(2x - x^2) = lim(c-->∞)∫(3 to c) dx/(2x - x^2) = lim(c-->∞)∫(c to 3) dx/x(2 - x)

Consider 1/x(2 - x) = A/x + B/(2 - x) = (A(2 - x) + Bx)/x(2 - x)

x: -A + B = 0; B = A = 1/2
1: 2A = 1; A = 1/2

lim(c-->∞)∫(3 to c) dx/x(2 - x) = lim(c-->∞)∫(3 to c) (1/2x + 1/2(2 - x)) dx = lim(c-->∞) (1/2 ln|x| - 1/2 ln|2 - x|)|(3 to c) = lim(c-->∞) (1/2 ln|c| - 1/2 ln|2 - c| - 1/2 ln|3| + 1/2 ln|1|) = lim(c-->∞) (1/2 ln|c/(2 - c)| - 1/2 ln|3|) = - 1/2 ln|3|

9) ∫(0 to ∞) e^(-x) sinx dx = lim(c-->∞) ∫(c to 0) e^(-x) sinx dx = lim(c-->∞) (-e^(-x)cosx|(c to 0) - ∫(c to 0)e^(-x)cosx dx) = lim(c-->∞) (-e^(-x)cosx|(c to 0) - (e^(-x)sinx|(c to 0) + ∫(c to 3)e^(-x)sinx dx))

f(x) = e^(-x)
f'(x) = -e^(-x)

g(x) = -cosx
g'(x) = sinx

--
f(x) = e^(-x)
f'(x) = -e^(-x)

g(x) = sinx
g'(x) = cosx

Consider 2∫(c to 0) e^(-x) sinx dx = -e^(-x)cosx|(c to 0) - e^(-x)sinx|(c to 3)
∫(c to 0) e^(-x) sinx dx = -1/2 e^(-x)cosx|(c to 0) - 1/2 e^(-x)sinx|(c to 0)


lim(c-->∞) (-1/2 e^(-x)cosx|(c to 3) - 1/2 e^(-x)sinx|(c to 0)) = lim(c-->∞) (-1/2 e^(-c)cos(c) + 1/2 e^(-0)cos(0) - 1/2 e^(-c)sin(c) + 1/2 e^(-0)sin(0)) = 1/2

10) ∫(3 to ∞) dx/(x^2 - 4) = lim(c-->∞) ∫(3 to c) dx/(x + 2)(x - 2)

Consider 1/(x + 2)(x - 2) = A/(x + 2) + B/(x - 2) = (B/(x + 2) + A(x - 2))/(x + 2)(x - 2)

x: B + A = 0; A = -B = -1/4
1: 2B - 2A = 1; 2B - 2(-B) = 1; B = 1/4

lim(c-->∞) ∫(3 to c) dx/(x + 2)(x - 2) = lim(c-->∞) ∫(3 to c) (1/4(x - 2) -1/4(x + 2)) dx = lim(c-->∞) (ln|x - 2|/4 - ln|x + 2|/4)|(3 to c) = lim(c-->∞) (ln|x - 2/x + 2|/4)|(3 to c) = lim(c-->∞) - 1/4 ln(1/5) = 1/4 ln(5)

11) ∫(0 to 1) x/(1 - x^2)^(1/2) dx = lim(c-->1-) ∫(0 to c) x/(1 - x^2)^(1/2) dx = lim(c-->1-) -(1 - x^2)^(1/2) |(0 to c) = lim(c-->1-) -(1 - c^2)^(1/2) = 1

I'm soooooooooooooooo tired! I didn't sleep well. Oh, my bed!

Are you ready for tomorrow?

Hehehe... I'm never ready for an exam. But the exam is always ready for me.

I didn't do my final fitness essay yet.

XD WTH That's so random.
You should have done it one month ago. Does the teacher accept it this late?

Nah, whatever. I'll tell her that I was too busy for taking care of Spirit for his injuries. XD

That's so lame. Well, you didn't any of the fitness assignment anyway.

You should have done it. Even thought it doesn't count much in your R-score but you must at least pass the class.

But it's Yoga, its intensity just kills me every time I go to the course. I can't possibly have the energy to do more at home.

You don't have to practice at home. Just write the essay and pretend that you have attain your goal.

You can't pretend that you have improved. She knows whether I have practiced or not only by looking at my works in class.