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Evaluate the integral.
$ \displaystyle \int \frac{1}{\sqrt{x + 1} + \sqrt{x}}\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 5
Strategy for Integration
Integration Techniques
Oregon State University
Harvey Mudd College
University of Michigan - Ann Arbor
Boston College
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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04:51
Find the Integral of \int …
Let's go ahead and multiply. It's happened bottom by the conjugated. So let's write this So this first part will stay in the same and then I'LL multiply its happened bottom by X plus one minus radical X. This is a common trick. I'm often simplifies things. So after multiplying viso in the numerator, you just multiplied by one. And then in the denominator, we have a difference of squares that something of this form. And when you multiply that out, you always get a square minus B square. So on our problem that she's going to be X plus one that's a swear and then minus piece where? Which is just X. So here, of course, I'm taking eh to be radical X plus one B equals radical X. Now let's just simplified this denominator here. That's just simplifies toe one. And then I could just bright this as X plus one radical X t X. So if this plus one is bothering you, go ahead and do a U substitution before he do the power rule. But in any case, after integrating this, we use the power rule. So that becomes X plus one to the three halfs power, and then you multiplied by two over three and then my tiss and then hear this becomes X to the three half power to over three plus e, and that's your final answer.
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