# Finding the Value of a Trigonometric Integral with Radical

• MHB
• karush
In summary, the given integral can be simplified to $I=\int _0^{2\pi }\sin{\frac x2}\,dx$ and after solving, the result is $4$.
karush
Gold Member
MHB
Evaluate
$$\int_0^{2\pi}\sqrt{\dfrac{1-\cos{x}}{2}}\,dx$$
ok my baby step is
$$\int _0^{2\pi }\frac{\sqrt{1-\cos \left(x\right)}}{\sqrt{2}}dx$$
then ?

W|A said the answer was 4

karush said:
Evaluate
$$\int_0^{2\pi}\sqrt{\dfrac{1-\cos{x}}{2}}\,dx$$
ok my baby step is
$$\int _0^{2\pi }\frac{\sqrt{1-\cos \left(x\right)}}{\sqrt{2}}dx$$
then ?

W|A said the answer was 4
$\cos x = 1 - 2\sin^2\frac x2$

$$I=\int _0^{2\pi }\frac{\sqrt{1-\cos \left(x\right)}}{\sqrt{2}}dx= \int _0^{2\pi }\dfrac{\sqrt{1-\left(1 - 2\sin^2\dfrac x2\right)}}{\sqrt{2}}dx= \int _0^{2\pi }\dfrac{\sqrt{2\sin^2{\dfrac x2}}}{\sqrt{2}}=\int _0^{2\pi }\sin{\frac x2}\,dx$$
then
$$I=-2\cos{\dfrac x2}\biggr| _0^{2\pi }=$$

ok I don't see this approaching 4

karush said:
$$I=\int _0^{2\pi }\frac{\sqrt{1-\cos \left(x\right)}}{\sqrt{2}}dx= \int _0^{2\pi }\dfrac{\sqrt{1-\left(1 - 2\sin^2\dfrac x2\right)}}{\sqrt{2}}dx= \int _0^{2\pi }\dfrac{\sqrt{2\sin^2{\dfrac x2}}}{\sqrt{2}}=\int _0^{2\pi }\sin{\frac x2}\,dx$$
then
$$I=-2\cos{\dfrac x2}\biggr| _0^{2\pi }=$$

ok I don't see this approaching 4
$$I=-2\cos{\dfrac x2}\biggr| _0^{2\pi }= -2(\cos\pi - \cos0) = -2(-1 - 1) = 4$$

## 1. What is a radical in trigonometric integrals?

A radical in trigonometric integrals refers to an expression that contains a root, such as a square root or cube root, of a variable or constant. It is often denoted by the symbol √.

## 2. How do you solve trigonometric integrals with radicals?

To solve trigonometric integrals with radicals, you can use techniques such as substitution, trigonometric identities, and integration by parts. It is also important to simplify the radical expression before integrating.

## 3. Can you give an example of a trigonometric integral with a radical?

One example of a trigonometric integral with a radical is ∫√(1 + sinx)dx. This integral can be solved using the substitution u = 1 + sinx, which simplifies the expression to ∫√udu. From there, you can use the power rule of integration to find the solution.

## 4. Are there any special cases when integrating trigonometric integrals with radicals?

Yes, there are some special cases when integrating trigonometric integrals with radicals. For example, if the radical expression contains only even powers, you can use the half-angle or double-angle identities to simplify the integral. Additionally, if the radical expression contains only odd powers, you can use the trigonometric substitution method to solve the integral.

## 5. Why are trigonometric integrals with radicals important in science?

Trigonometric integrals with radicals are important in science because they can be used to solve a variety of problems in fields such as physics, engineering, and mathematics. They can also help in understanding and modeling natural phenomena, such as wave motion and oscillations, which are often described using trigonometric functions with radicals.

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