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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int x^3 \arcsin (x^2)\ dx $
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Calculus 2 / BC
Chapter 7
Techniques of Integration
Section 6
Integration Using Tables and Computer Algebra Systems
Integration Techniques
Campbell University
Oregon State University
Harvey Mudd College
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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Use the Table of Integrals…
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Use the Table of Integral…
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Okay, so this question wants us to compute this integral. So what we need to dio is find a substitution that turns this into one that we can find its value easily from our table. So let's pick you to be the arguments of the arc sine so X squared. So if we take to you, we get two x d x, so 1/2 d'you is equivalent to x t X. So then let's write this integral in a slightly different form so we can see what happens here. So we see that X squared is you and X d X is 1/2 to you. So this becomes 1/2 times the integral of you Sign in verse of you, do you? So now from there we can use the following formula from our table. So this tells us what the integral of a signed inverse times you is. So so are integral is 1/2 times this thing. So now all we have to do is distribute the two to both terms and replace you with X squared. So this gives us two X to the fourth because you squared squared. We're sorry. Ex word squares except the fourth minus one over a sine inverse of X squared plus X squared square root one minus X to the fourth all over Hey plus C, and that's our final answer.
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