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Evaluate the integral.
$ \displaystyle \int \frac{xe^{2x}}{(1 + 2x)^2} dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 1
Integration by Parts
Integration Techniques
Missouri State University
Baylor University
Idaho State University
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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The problem is violated the integral x times e to 2 x, over 1 plus 2 x into the square for this problem. We we can use my third integration by powers. The formula is integral of: u: v from d x is equal to u times, v minus the integral of prime v d x. Now for our problem, we can write. U is equal to x times e to 2 x and prime is equal to 1 over 1 plus 2 times x square. Then you problem is we use product rule here, so this is e to 2 x plus x times e to 2 x times 2. This is 1 plus 2 x times e to 2 x and v is equal to negative 1 half times 1 over 1 plus 2 x now use this formula. Integral of this function is equal to u times. This is negative x over 2 times 1 plus 2 x times e to 2 x. Minus you prime times we. This is minus integral of 1 plus 2 x times e to 2 x times negative 1, half 1 over 1 plus 2 x x simplified this function. We have this is equal to negative x into 2 x over 2 times 1 plus 2 x minus this plus il half integral of into 2 x x. The answer is, this is equal to negative x times e to 2 x over 2 times 1 plus 2 x plus 1 force into 2 x and plus constant number c.
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