Just stucked in Integration

I was generalising the result of moment of inertia of ellipse parallel to $xy$ plane having equation \[\rightarrow \textcolor{Blue}{\Large \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 }\]

I reached this $2$ integrals and after that can't able to solve $\Rightarrow \textcolor{#20A900}{\Large\int_{0}^{\frac{π}{2}} \sqrt{1+\alpha \cos^{2}\theta}d\theta}$ $\Rightarrow \textcolor{#20A900}{ \Large \int_{0}^{\frac{π}{2}}\sin ^{2}\theta \sqrt{1+\beta \cos ^{2}\theta}d\theta}$
$\alpha, \beta$ are constants. I will be very grateful if anyone will help me.

#Calculus

Easy Math Editor

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Comments

You should check out Elliptic Integrals....

Aaghaz Mahajan - 1 year ago

Like Aaghaz said: There is no elementary closed form for this integral. Elliptic integrals

Pi Han Goh - 1 year ago

Please help @Karan Chatrath @Steven Chase @Mark Hennings @Guilherme Neidu @Pi Han Goh @Alak Bhattacharya @Aaghaz Mahajan @Naren Bhandari

A Former Brilliant Member - 1 year 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}`	$\boxed{123}$