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Problem 565

# Express $\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right) \text { du in terms of incomplete elliptic }}$$integrals where$$0 \leq \mathrm{x} \leq(\pi / 6)$.

Expert verified
The given integral can be expressed in terms of an incomplete elliptic integral of the second kind as $\mathrm{x} E(\mathrm{u}, 2), \quad 0 \leq \mathrm{u} \leq (\pi / 6).$
See the step by step solution

## Step 1: Recall the definition of an incomplete elliptic integral of the second kind

An incomplete elliptic integral of the second kind, denoted as $$E(\phi, k)$$, is defined as: $E(\phi , k) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2 \theta} \mathrm{ d \theta}.$

## Step 2: Compare the given integral to the elliptic integral definition

Given integral is: $\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right)} \mathrm{du}.$ Comparing it with the definition of an elliptic integral of the second kind: $E(\phi , k) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2 \theta} \mathrm{ d \theta},$ We can notice the similarities as both the integrals have the same integral form. Now, in order to rewrite the given integral in terms of elliptic integrals, we need to identify $$\phi$$ and $$k$$.

## Step 3: Identify $$\phi$$ and $$k$$ from the given integral

The given integral is: $\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right)} \mathrm{du}.$ Comparing it with the definition of an elliptic integral of the second kind $E(\phi , k) = \int_{0}^{\phi} \sqrt{1 - k^2 \sin^2 \theta} \mathrm{ d \theta},$ we can make the following identifications: 1. $$\phi = \mathrm{u}$$ 2. $$k^2 = 4$$ With these identifications, we have $$k = 2$$.

## Step 4: Rewrite the given integral in terms of elliptic integrals

Now that we have identified $$\phi$$ and $$k$$, we can rewrite the given integral in terms of the incomplete elliptic integral of the second kind. The given integral is: $\mathrm{x} \int_{0} \sqrt{\left(1-4 \sin ^{2} \mathrm{u}\right)} \mathrm{du}.$ Using our identifications from Step 3, we can express the integral as: $\mathrm{x} E(\mathrm{u}, 2).$ Since $$0 \leq \mathrm{x} \leq (\pi / 6)$$, we can conclude: $0 \leq \mathrm{u} \leq (\pi / 6).$ So, the expression of the given integral in terms of an incomplete elliptic integral of the second kind is: $\mathrm{x} E(\mathrm{u}, 2), \quad 0 \leq \mathrm{u} \leq (\pi / 6).$

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