MHB Which is Larger: \(\int_{0}^{\pi} e^{\sin^2 x}\,dx\) or \(\frac{3\pi}{2}\)?

[anemone]

Which of the following is the bigger?

$$\int_{0}^{\pi} e^{\sin^2 x}\,dx$$ and $\dfrac{3\pi}{2}$

 

[Opalg]

Which of the following is the bigger?

$$\int_{0}^{\pi} e^{\sin^2 x}\,dx$$ and $\dfrac{3\pi}{2}$

[sp]
Replace $x$ by $\sin^2x$ in the inequality $e^x > 1+x$ (which holds for all $x$ except $x=0$) to see that $$\int_{0}^{\pi} e^{\sin^2 x}\,dx > \int_0^\pi (1 + \sin^2x)\,dx = \pi + \frac\pi2 = \frac{3\pi}2.$$[/sp]

 

[Nono713]

[sp]
Replace $x$ by $\sin^2x$ in the inequality $e^x > 1+x$ (which holds for all $x$ except $x=0$) to see that $$\int_{0}^{\pi} e^{\sin^2 x}\,dx > \int_0^\pi (1 + \sin^2x)\,dx = 1 + \frac\pi2 = \frac{3\pi}2.$$[/sp]

Spoiler

Small typo, it should read $\pi + \frac{\pi}{2}$ :)

 

[anemone]

[sp]
Replace $x$ by $\sin^2x$ in the inequality $e^x > 1+x$ (which holds for all $x$ except $x=0$) to see that $$\int_{0}^{\pi} e^{\sin^2 x}\,dx > \int_0^\pi (1 + \sin^2x)\,dx = \pi + \frac\pi2 = \frac{3\pi}2.$$[/sp]

Well done, Opalg!(Happy) And thanks for participating!;)