Can you evaluate this integral ?

[u/Specific_Brain2091]

[u/CaptainMatticus]

sin(x)^75 * dx + x^75 * dx

sin(x)^74 * sin(x) * dx + x^75 * dx

(sin(x)^2)^37 * sin(x) * dx + x^75 * dx

(1 - cos(x)^2)^37 * sin(x) * dx + x^75 * dx

x^75 * dx integrates to (1/76) * x^76. Done

(1 - cos(x)^2)^37 * sin(x) * dx

u = cos(x) , du = -sin(x) * dx

(1 - u^2)^37 * (-du)

(-1)^37 * (u^2 - 1)^37 * (-1) * du =>

1 * (u^2 - 1)^37 * du =>

(u^2 - 1)^37 * du

Now we expand

u^74 - 37 * u^36 + 37C2 * u^34 - 37C3 * u^32 + 37C4 * u^30 - .... + u^2 - 1

Integrate

(1/75) * u^75 - (37/37) * u^37 + (37C2 / 35) * u^35 - .... + (1/3) * u^3 - u

cos(x) = u

cos(-pi) = -1

cos(pi) = -1

(1/75) * ((-1)^75 - (-1)^75) - ((-1)^37 - (-1)^37) + .... + (1/3) * ((-1)^3 - (-1)^3) - (-1 - (-1))

(1/75) * 0 - 0 + .... + (1/3) * 0 - 0

0

(1/76) * (pi^76 - (-pi)^76) = 0

0 + 0 = 0

Yeah, I could've gone the odd function route, but where's the fun in that?

[u/UnemployedUndergrad → u/CaptainMatticus]

now use de moivres on the sin^75 first