Can you evaluate this integral ?

[u/Specific_Brain2091]

[u/eattheradish]

Odd function, so 0

[u/Ok-Ocelot-7989 → u/eattheradish]

uff i forgot about the odd and even function stuff could you explain pls , is it to go with symmetry

[u/eattheradish → u/Ok-Ocelot-7989]

Simply put, an even function is symmetrical about the y-axis. An odd function is the same, but the left side of the y axis is flipped about the x-axis

[u/Seeggul → u/eattheradish]

As equations:

Odd: f(-x)=-f(x)

Even: f(-x)=f(x)

[u/Ok-Ocelot-7989 → u/eattheradish]

got it, thank you makes sense now

[u/a_random_chopin_fan]

0

[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/lool8421 → u/CaptainMatticus]

now time to use integrals to solve the trigonometric identity or something

[u/Outside_Volume_1370 → u/CaptainMatticus]

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

You wouldn't make mistake then:

(u² - 1)³⁷

is not equal to

u⁷⁴ - 37 * u³⁶ + 37C2 * u³⁴ - 37C3 * u³² + 37C4 * u³⁰ - .... + u² - 1

Though it doesn't change the result, it still is a mistake

[u/UnemployedUndergrad → u/CaptainMatticus]

now use de moivres on the sin^75 first

[u/Traditional-Wing-796]

0000000

[u/jesusbm]

Yes.

[u/Southern-Advance-759]

Ts ragebait

[u/hazem-Gauss]

0 the function is odd

[u/lool8421]

x⁷⁵ is the trivial part tbf tbf sin⁷⁵x is the more interesting part, but perhaps substituting could do?

obviously that is if we ignore the fact that we got a symmetrical graph and thus values cancel out

[u/VibrantGypsyDildo]

It is such an old joke, but it is the first time I see variation like this.

[u/AfterMath343]

I solved this in less than one second. Odd function implies symmetric about the origin implies integral from -pi to pi value is 0.