Fool's problem of the day,
Wonderful Integrals:
1. Evaluate: \( \huge \int \frac{\sin x}{\sqrt{1+\sin x}} \)
2. Prove \( \huge \int \limits_{0}^{\frac \pi 2}\frac{x}{\sin x+\cos x}dx = \frac{\pi}{2\sqrt{2}} \ln (\sqrt{2} +1) \)
3. Prove \( \int \limits_{0} ^1 \sqrt{(1+x)(1+x^3)} \; dx \le \sqrt{\frac{15}{8}} \)
4. Let \( a,b, c \) be non-zero real numbers, such that \( \int \limits_{0} ^1 (1+ \cos ^8 x)(ax^2 bx +c) \; dx = \int \limits_{0} ^2 (1+ \cos ^8 x)(ax^2 bx +c) \; dx \). Then show that the quadratic equation \( ax^2 +bx+c=0\) has one root in \( (1,2) \)
[Solved by @Mr.Math]
PS: First two are probably very easy! ;)
@TuringTest Time for you solve my problem of the day! Between I would only accept all the solutions from you ;)
Good luck!

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How can integrals be wonderful -_-

They are once you know them -_=

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