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Evaluate the integral.
$ \displaystyle \int te^{-3t} dt $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 1
Integration by Parts
Integration Techniques
Missouri State University
Harvey Mudd College
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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Hello. So we're trying to evaluate into girl of t E to a native three tt This is a standard integration by parts problem. So you have to figure out a you and a devi to decompose the problem with. Usually you want to find a you that you khun keep differentiating until you get to like one. So that you, Khun, basically get rid of it. And the DV needs to be something that you can integrate visioning your conventional methods. So in this case, it's easy to see the U equals T and D vehicles even Eva three t I say easy because I've done this so many times. It comes with experience. Do you would then be equal to DT? The is equal to e to the negative one third e to the negative three t So what happens next? Put it together. I color coded it. So it's easier to see I put the constants in France. It's easier to work with afterwards, So you have equals to you tempt me which is they won thirty to the in the eternity of three t minus then ago VD you So that's minus ah native one third in a girl of eighteen eighty three T d. T. So you get a plus one third in the next line. Um, and then integrating into ninety three t by itself would just would he also give you another native one third term in a man we get eked native one thirty, eatin a three T minus one ninth eater native three t plus C.
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