Find the general indefinite integral.
$ \displaystyle \int (\sin x + \sinh x)\, dx $
$$-\cos x+\cosh x+C$$
for this problem. We've been asked to find the general indefinite integral for a given expression sine of X plus the hyperbolic sine of X. Now a couple things to keep in mind. Here first, any time we're taking an indefinite integral, we know we're going to be finishing our answer with a plus C. That's because think about going the other direction when you take a derivative derivative of a constant, just become zero. So if you're going to go back, you need to recognize the fact that you could have had a constant there that you no longer see when you take the derivative. So taking the integral we're always going to add that plus C on the end. The other thing is because this is a a sum of two functions. We can actually break this up. I can say the thesis, the integral of sine of X dx, plus the integral of the hyperbolic sine of X dx. So if you have a sum of two functions, you can break that up and take the integral of each piece independently and add them together the same. If it was a difference, I could take each piece take the integral independently and take the difference of those integral. So let's take a look at these first. I have sine of X. What's the integral of sine of X? The integral of sine of X is negative co sign of X plus. See? Remember, we have to add that plus C now, what about the hyperbolic sine? While the derivative are the integral of hyperbolic, sine is hyperbolic co sign and again with a plus C. Now we're going to do one more line just to finish simplifying this I have two plus sees. These are both unknown. Constance, an unknown constant plus an unknown constant is just an unknown constant. So I don't have to have a C for each individual piece here. We can put this together with a single C at the end, and that is our general indefinite Integral