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Evaluate the integral.
$ \displaystyle \int x \cosh ax dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 1
Integration by Parts
Integration Techniques
Missouri State University
Campbell University
Boston College
Lectures
01:53
In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.
27:53
In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.
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Calculate the integral.
The problem is evaluated the integral x times coach, a x d x, the first, the coach a x. This is, by definition, this is a 2 x plus e to negative x over 2 point, and we have the derivative of coach a x coach x. So this this is the definition of coach x. The coach x derivative of this 1 is equal to sine x. This is 2 x minus e to 2 x over 2. Now for this problem, we were used the method of integration by parts. The formula is integral of u v. Prime dx is equal to: u times, v minus the integral of u prime times v d x for our problem. We can, let? U is equal to x and the prime is equal to poach 8 times x. Then? U, prime, is equal to 1, and v is equal to 1 over 8 times inch a x now integral x times, coach x x. This is equal to u times v, so this is 1 over a times x, times c n a x and minus integral of prime timotes is 1 a cinch a x x. This is equal to 1 over 8 times x, times x, minus 1 over a square cosine, the coach x and plus the constant number. This is the answer:
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