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In my calculus class in university, we learned how to use substitution to solve integrals. But then on the quiz we got this question:

Find the numerical value of the following integral:

$$\int_0^\frac{\pi}{2} \frac{{\sin(x)}^{22222222}}{{\sin(x)}^{22222222} + {\cos(x)}^{22222222}} dx$$

Hint: Use the substitution $u = \frac{\pi}{2} - x$.

Second hint: You cannot find an elementary anti-derivative of this.

Give your answer with 3 decimal digits.

Recommended time: 10 minutes

I could not understand it, even when the teacher explained it after the quiz. He was saying that it is the same for every power of the form $2n$ where $n$ is a natural number.

What should I do to solve this?

DMcMor
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Hatchi Roku
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    https://math.stackexchange.com/questions/439851/evaluate-the-integral-int-frac-pi2-0-frac-sin3x-sin3x-cos3x/439856#439856 – lab bhattacharjee Dec 10 '20 at 14:51
  • Thanks man, I see the key of the solution and what we're trying to do here. But I didn't understand why $\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$ – Hatchi Roku Dec 10 '20 at 14:58
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    @AmeerTaweel try the substitution $y=a+b-x$ and use the fact that $\int_a^b f(x),dx = \int _a^b f(y),dy$. – Integrand Dec 10 '20 at 16:42
  • @overrated but then I get dx = -dy, so the integral becomes $\int_a^b f(x) dx = \int_b^a f(y) dy$ – Hatchi Roku Dec 10 '20 at 19:32
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    Keep track of the negative and switch the limits of integration. – Integrand Dec 10 '20 at 19:50
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    Thank you, now I see my mistake, the limits of integration are first switched and then the negative sign from dx = -dy returns them to the original order. – Hatchi Roku Dec 10 '20 at 19:54

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