Welcome! Today we'll solve the integral of square root of 1 plus sine x. This is a classic integral that requires clever use of trigonometric identities. Let's start by examining the function we need to integrate.
The key insight is to rewrite 1 plus sine x using trigonometric identities. We use the Pythagorean identity: 1 equals sine squared of x over 2 plus cosine squared of x over 2. We also use the double angle formula: sine x equals 2 sine of x over 2 times cosine of x over 2. Combining these, we get a perfect square under the radical.
Now we simplify the expression inside the absolute value. We factor out square root of 2 and recognize that one over square root of 2 equals both cosine and sine of pi over 4. Using the sine addition formula, we can write this as square root of 2 times sine of x over 2 plus pi over 4. This transforms our integral into the absolute value of a sine function.
Now we must handle the absolute value. The sine function is positive in certain intervals and negative in others. For x in the interval from 4k pi minus pi over 2 to 4k pi plus 3 pi over 2, the sine is non-negative. For x in the interval from 4k pi plus 3 pi over 2 to 4 times k plus 1 pi minus pi over 2, the sine is negative. This gives us two different integrals to evaluate.
Here's our final solution. The integral of square root of 1 plus sine x is a piecewise function. For the first interval, the result is 2 times sine of x over 2 minus cosine of x over 2 plus a constant. For the second interval, it's 2 times cosine of x over 2 minus sine of x over 2 plus another constant. The constants are related to ensure continuity. Alternatively, we can express this as negative 2 square root of 2 times the absolute value of cosine of x over 2 plus pi over 4.