Integration of √sin(x)

Introduction:
The integration of the square root of sin(x), denoted as √sin(x), involves finding an antiderivative or integral of the function with respect to the variable x. This process allows us to determine the area under the curve of the function or find a function that yields the original function when differentiated.

Approach:
To integrate the function √sin(x), we can use a combination of trigonometric identities, substitution, and integration techniques.

Trigonometric Identity:
To simplify the integration process, we can use the following trigonometric identity:
sin^2(x) = (1 - cos(2x))/2

Substitution:
We can substitute u = cos(x), du = -sin(x)dx to rewrite the function as:
√sin(x) = √(1 - cos^2(x)) = √(1 - u^2)

Integration Technique:
We can now perform the integration using the substitution method and the trigonometric identity. Let's break down the process into steps:

1. Substitute u = cos(x) and du = -sin(x)dx.
2. Rewrite the function in terms of u: √(1 - u^2).
3. Apply the trigonometric identity sin^2(x) = (1 - cos(2x))/2 to simplify the function.
4. Replace sin(x) with -du in the identity: sin^2(x) = (1 - cos(2x))/2 = (1 + u^2)/2.
5. Rewrite the function in terms of u: √(1 - u^2) = √[1 - (-du)^2] = √(1 + u^2).
6. The function is now simplified to √(1 + u^2).

Integration of √(1 + u^2):
The integral of √(1 + u^2) can be evaluated using trigonometric substitution, specifically by letting u = tan(theta). By doing so, we get:

∫√(1 + u^2) du = ∫√(1 + tan^2(theta)) sec^2(theta) d(theta)
= ∫sec^3(theta) d(theta)

The integral of sec^3(theta) can be computed using integration by parts or other techniques like tabular integration.

Conclusion:
The integration of √sin(x) involves using trigonometric identities, substitution, and integration techniques. By substituting u = cos(x) and simplifying the function, we can convert the original problem into the integration of √(1 + u^2). This new integral can be evaluated using trigonometric substitution and further techniques like integration by parts.

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