Which one is Bigger?

Which of these numbers is larger?

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

#Calculus

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 5 years, 1 month 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 $1.5\pi$

Hamza A - 5 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$