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Evaluate the integral.
$ \displaystyle \int \ln \sqrt{x} dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 1
Integration by Parts
Integration Techniques
Harvey Mudd College
University of Michigan - Ann Arbor
Idaho State 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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Evaluate the integral.…
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Evaluate the indefinite in…
The problem is, you violated the integral integral of n root of x d x for this problem. We cannot use the method of integration by powers, so the formula is integral of u v. Prim dx is equal to: u times, v minus integral of? U? Prime v d x for this problem we can not? U is equal to n rotonda prime is equal to 1. It then prime, is equal to here. We use the change rule. This is motive x, 12 x times. Integral of double x is 1 half times 1. Over root of x, so this is 1 over 2 x in v- is equal to x. Over have integral of x. Dx is equal to u times v, so this is x times n root of x minus integral of prim times. So this is 1 half the answer is x times: l n, o t of x, minus 1, half times x and plus the constant number c.
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