Today we'll solve the integral of square root of 1 plus sine x. This is a challenging integral that requires clever use of trigonometric identities. Let's start by examining the function we need to integrate.
To solve this integral, we need to use trigonometric identities. First, we recall that 1 equals sine squared of x over 2 plus cosine squared of x over 2. We also use the double angle identity: sine x equals 2 sine of x over 2 times cosine of x over 2. By substituting these identities, we can rewrite the expression under the square root.
Now comes the key insight. The expression we obtained is actually a perfect square. We can recognize that sine squared of x over 2 plus cosine squared of x over 2 plus 2 sine of x over 2 times cosine of x over 2 follows the pattern a squared plus 2ab plus b squared, which equals a plus b squared. Therefore, 1 plus sine x equals sine of x over 2 plus cosine of x over 2, all squared.
Now we evaluate the integral. For the general indefinite integral, we assume the expression inside the absolute value is positive. We integrate each term separately: the integral of sine of x over 2 gives negative 2 cosine of x over 2, and the integral of cosine of x over 2 gives 2 sine of x over 2. Combining these results, we get our final answer: 2 sine of x over 2 minus 2 cosine of x over 2 plus the constant of integration.
Let's summarize our solution. We successfully integrated the square root of 1 plus sine x by recognizing that the expression under the square root is a perfect square. The key insight was using trigonometric identities to rewrite 1 plus sine x as sine of x over 2 plus cosine of x over 2, all squared. This allowed us to simplify the integral and obtain our final answer: 2 sine of x over 2 minus 2 cosine of x over 2 plus C. This technique of recognizing perfect squares is a powerful tool in integration.