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Which one is Bigger?

Which of these numbers is larger?

\[\large \int_0^\pi e^{\sin^2 x} \, dx \qquad \text{ OR } \qquad 1.5 \pi \ ? \]

Note by Leo X
6 months, 4 weeks ago

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Note that \( e^x = 1 + x + \frac{x^2}{2!} + ... > 1 + x \) for \( x > 0 \).

Therefore, \( \int_0^\pi e^{\sin^2 x} dx > \int_0^\pi 1 + \sin^2x dx = \frac{3\pi}{2} \) Siddhartha Srivastava · 6 months, 3 weeks ago

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the integral is equal to \(\pi \sqrt { e } { I }_{ 0 }\left( \frac { 1 }{ 2 } \right) \) where \(I_n\) is the modified Bessel function of the first kind,the answer comes out to be \(\approx5.5\) which is greater than \(1.5\pi\) Hummus A · 6 months, 3 weeks ago

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