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Evaluate the integral.

$ \displaystyle \int \sin x cos \left (\frac{1}{2} x \right) dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Campbell University

Baylor University

University of Michigan - Ann Arbor

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…

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this problem is from Chapter seven, Section two. Problem number eighteen in the book Capitalist Early Transcendental Sze eighth Edition by James Door And here we have a indefinite noble of sine x times co sign of X over too. So the first thing we can do here is rewrite the sign of eggs. So let's let's write that as sign of two times eggs over too, and then applying the double angle formula over here on the right, we have two sign eggs over too times co signing books over to and we have this extra factory coastline of exit too. So we could simplify this by pulling on it too, combining these co signs. So at this point, we were ready to use u substitution. And here we should take you two be co sign the checks over to. So if we do this, then do you is negative sign except for two. And then using the chain rule left the multiplied by one half and then the ex which we could simplify as negative too. Do you this sign XO over too dx. So if we apply this u substitution, we have a two times negative too So we have a negative for in a girl and then you square, do you, which we can integrate using the pot rule, we get a negative for you, Cube over three plus e and then coming back to our use up. We can replace you with co signing books over to to get our final answer, and that's our answer.

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