Use the Table of Integrals on the Reference Pages to...

Key Concepts

Chain Rule in Reverse (u-Substitution)

The reverse application of the chain rule, commonly executed through u-substitution in integration, is crucial when dealing with composite functions. When an integral involves a function nested inside another (for example, an inverse trigonometric function applied to a rational function), recognizing the derivative of the inner function enables the use of substitution to simplify the integration process.

Table of Integrals

A table of integrals is a compiled reference that lists standard antiderivatives for a variety of integrals. It is used to quickly find the antiderivative of functions that match or can be manipulated to match one of the standard forms. In problems like the given one, the table can be used to recognize the antiderivative of an integrand that might involve inverse trigonometric functions or other complex compositions, easing the evaluation process.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine, are functions that reverse the action of the standard trigonometric functions. They appear in integration, often leading to integrals that cannot be expressed in terms of elementary functions in any other form. A thorough understanding of their properties and derivatives is essential when evaluating integrals that include these functions.

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. This technique often involves identifying an inner function whose derivative is also present (or can be manipulated to be present) in the integrand. In integrals that involve compositions of functions, such as those with inverse trigonometric functions inside another function, substitution can help convert the integral into a standard form listed in a table of integrals.

Transcript

00:01 Okay, we want to solve the indefinite integral that is of the function.

00:06 Inverse cosine of x the minus 2 over x cubed dx.

00:11 And to do this, we'll be using u substitution so that we can use this following, this following integral solution from our given integral table, which is entry 88 in the given textbook, or again, there are many integral table sheets online.

00:30 So by using use substitution, we will reduce the.

00:32 This integral such that we're only finding the integral of arc cosine of u, which is equal to u times arc cosine of u minus root 1 minus u squared plus c.

00:43 So let's begin with our u substitution and say that our u is just equal to our x to the minus 2.

00:54 Because this way we should be able to cancel out this x cubed such that we're only dealing with our cosine, which is the goal, because we want to use this integral.

01:04 Formula.

01:06 So if u is equal to x to the minus two, this means that d u is equal to negative two, because again we do the power rule here, minus two, then subtract one from the exponent.

01:19 So we get minus two times x to the minus three, dx.

01:26 So as we can see, we have an x to the minus three and a dx already in our original integral.

01:33 So this means we have to bring the minus two over to the d u side so we can make a proper substitution.

01:38 So this means we have a negative 1 1 half du equal to x minus 3 dx so i'll make the substitution we can write out what this looks like so this means our integral is now multiplied by a factor of minus 1 half such that we can then write that we have a cosine or inverse cosine of u and the x the 1 over x cubed and the dx gets subbed with our negative 1 half du so this just means we have du because we already have our factor of negative 1 half...

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