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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
8 months, 2 weeks ago

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

Therefore, $$\int_0^\pi e^{\sin^2 x} dx > \int_0^\pi 1 + \sin^2x dx = \frac{3\pi}{2}$$ · 8 months, 2 weeks ago

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 $$1.5\pi$$ · 8 months, 2 weeks ago