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Find the general indefinite integral.

$ \displaystyle \int (\sin x + \sinh x)\, dx $

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00:54

Frank Lin

00:40

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 4

Indefinite Integrals and the Net Change Theorem

Integration

Campbell University

Harvey Mudd College

University of Nottingham

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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Find the general indefinit…

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Find the indefinite integr…

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Evaluate the indefinite in…

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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

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