Integration of (sinx)½ from 0 to π/2:

To find the integral of the function (sinx)½ from 0 to π/2, we can use the definite integral formula and evaluate it step by step.

Step 1: Identify the function and limits of integration:
The function to be integrated is (sinx)½, and the lower limit is 0 while the upper limit is π/2.

Step 2: Apply the power rule for integration:
The power rule for integration states that when integrating a function of the form xn, the result is (1/(n+1)) * xn+1. In this case, the power is ½, so we can apply the power rule to integrate (sinx)½.

Step 3: Integrate the function:
Using the power rule, we can integrate (sinx)½ as follows:
∫(sinx)½ dx = ∫sinx^(1/2) dx
= (2/3) * sinx^(3/2) + C

Step 4: Evaluate the definite integral:
To find the definite integral with limits of integration, we substitute the upper and lower limits into the integrated function.

Using the definite integral notation, we have:
∫[0,π/2] (sinx)½ dx = [(2/3) * sinx^(3/2)] evaluated from 0 to π/2

Step 5: Substitute the values and calculate:
Substituting the upper and lower limits into the integrated function, we have:
[(2/3) * sin(π/2)^(3/2)] - [(2/3) * sin(0)^(3/2)]

Simplifying further:
[(2/3) * 1^(3/2)] - [(2/3) * 0^(3/2)]
= (2/3) - 0
= 2/3

Step 6: Final Answer:
Therefore, the definite integral of (sinx)½ from 0 to π/2 is 2/3.

In summary, by applying the power rule for integration, we found that the definite integral of (sinx)½ from 0 to π/2 is 2/3.