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Evaluate the integral.
$ \displaystyle \int_0^\pi x \sin x \cos x dx $
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Calculus 2 / BC
Techniques of Integration
Integration by Parts
Harvey Mudd College
University of Michigan - Ann Arbor
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.
In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.
Evaluate the integral.…
Evaluate the integral.
to evaluate the definite integral from 0 to Pi of X times Sine of X times Cosine of X. Dx. We know that sign of two X. This is equal to to sign up X times Cosine of X. And so 1/2 sine of two x. This is just sine of X times go sign of X. Therefore we can read our integral into the integral from 0 to Pi. Ah X times 1/2. sign up two X. Dx and we can Right the constant out. We have 1/2 Integral from 0 to Pi of X times sine of two X dx. From here we want to apply integration by parts. We let U equal to X. And DV Sign of two x. DX so that the U. S. D. X. And we would be Negative co sign of two x over two. So from here we have one half times U times V. That's negative x times go sign up to X over two minus the integral of V times do you? That's negative co sign of two X. Over to the X. Simplifying that we have one half times negative X times go Sign up to X over two Plus we have 1/2 integral of course and have two X. That's just sign of two X over two. And this will be evaluated from zero to pi. And so from here we have negative X. Co sign up to X over four Plus sign of two x Over eight. This evaluated from 0 to Pi And if excess spy we have negative pi close enough to buy Over four plus, sign up to buy over eight. This minus when X0, we have zero Plus sine of 0/8. This is just zero and so we have also. This one is zero and co sign of two pi is one. So the value of the integral would be Negative by over four.
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