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Evaluate the integral.
$ \displaystyle \int_0^3 \frac{x}{\sqrt{36 - x^2}}\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 3
Trigonometric Substitution
Integration Techniques
Missouri State University
Campbell 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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here we have a definite nibble from Sierra three X over the square root of thirty six, minus x squared. Since our denominator is of the form ace flirt minus X squared. We should use the tricks of of the form X equals a sign data. Here we see that a equal six. So we should take X to be six in data then D X is six co signed data data. So notice here we have. Ah, definitely a rule. So we have two options on how to proceed. The first option is to find the new limits of integration in terms of data. And if you go that road, you won't have to draw the triangle and you won't have to back substitute because there won't be any need to go back into the variable X. So I prefer that option. The second option is to draw, evaluate, integral, draw the triangle back, substitute back into the variable X, and then you could plug in zero and three is your ex values into the anti derivatives, so that will require the triangle if you want to go that way. So in this problem, let's see if we could find the new limits of integration. So here noticed that from the textbook. When we make this trick substitution, we always take fader to be between negative power too. And powerful too. This will help you find data. So the lower limit before was zero. So if we plug in X equals zero into this equation up here, excuse me into this equation down here, we could solve her data. So we have zero equals six i data. So this means signed data equals zero and sense data has to fly in this interval. There's only one solution, and that's exactly data equal zero. So that's it. One of the new lower limit of integration. The upper limit was ex people story. So plugging X equal three into this equation again, we have three equals. Six idea, so scientific is a half. And then, in this interval from negative Piper Tuna Piper to the only time data is a sign Data's a half is when data is five or six. So that's the new upper limit. So here one more observation. We have this radical in the denominators. Let's quote and simplify that First thirty six minus X squared is thirty six minus thirty six Sign square. We can pull out a thirty six out of the radical that become six. Then we have one minus sign square. So this is equal to six square rule of co signs where which is equal to six co signed data. Come over here and write that six co signed data so that that's our new denominator. So now we have in a rule zero to power six x and the numerator. So it's six nine data DX, which was six coastline data. And then we have six co signed data on the bottom. That's coming from the denominator that we simplified. We could go out and cancel off these six co signs. Pull out the six. So we have six Integral Sierra five or six signed data. So this is negative. Six. Let me write it this way. Six anti derivative of sinus negative co sign and then we have our end points zero, five or six. So plugging in those end points. We have negative six times radical three over too plus six, which we can write as negative. Three. Room three plus six. And there's our answer
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