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Evaluate the integral.
$ \displaystyle \int \frac{4^x + 10^x}{2^x}\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 5
Strategy for Integration
Integration Techniques
University of Michigan - Ann Arbor
University of Nottingham
Idaho State University
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.
00:49
04:44
Evaluate the integral.…
03:36
05:38
for this in a girl. Let's just go ahead and briefest into two fractions. So we have four x over to the ex. Tend to the X over two to the ex. And then here we can use the fact that eighteen eggs over beat in X is a over B to the X that she's using laws of experiments. So here will just have four or three to the ex ten over to to the ex. Now let's simplify these. That's just two decks and then fact to the ex, and we could have a formula for these exponential inaugurals. This is just eight of the X over the natural log of a plus he and here is a verification of this. If you forgot this fact, you could just check that. This is true, because if you differentiate this over here, you will get eight X over Helen A. But then you also have to multiply by Ellen A. When you differentiate the index so those would cancel. The Z will go to zero and you will be left over with the inside grand. So here, using that formula twice once for the two and then once for the five, let's add that constant of integration, see, And that's your final answer
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