Methods of Integration Show that the following results are equivalent. Integration by tables: ∫√

step 1 And basic integration roots, we want to verify giving an integral. This is an integral square ro

Methods of Integration Show that the following results are...

Key Concepts

Indefinite Integration

This concept involves finding antiderivatives of functions, meaning functions whose derivatives return the original integrand. In the context of integration of functions like sqrt(x^2 + 1), it is important to understand that there are multiple forms of antiderivative expressions that differ only by an arbitrary constant.

Inverse Hyperbolic Functions

Inverse hyperbolic functions, such as the inverse hyperbolic sine (arcsinh), provide an alternative way to express certain logarithmic forms. Recognizing that arcsinh(x) is equivalent to ln(x + sqrt(x^2 + 1)) is key to understanding the equivalence between different antiderivative expressions.

Equivalence of Antiderivatives

This principle states that any two antiderivatives of a function differ only by a constant. Demonstrating that different forms of the integral, whether expressed in logarithmic terms or using inverse hyperbolic functions, are equivalent reinforces the idea that they represent the same family of solutions.

Integration by Tables

Integration by tables is a method where one uses precompiled tables of integrals to match the integrand with a known formula, thereby simplifying the process. It is an efficient method for solving integrals that might otherwise require more intricate substitution or manipulation techniques.

Transcript

00:05 And basic integration roots, we want to verify giving an integral.

00:16 This is an integral square root of x squared plus 1 the x.

00:26 So we want to verify if we solve, i mean using integration tables and if we use computer algebra, whether they are equivalent.

00:37 So let's look at the integration.

00:41 So let's say my function here.

00:45 So let's let f of x be equal to the initial function that we have.

01:00 That is 1 over 2 of what x square root of x squared plus 1.

01:14 Then i have flat my lane of x plus 1 x plus square root square root of x squared plus 1 x squared plus 1 bucket close plus c now let's check with it so what we want to do is let's take the derivative of this function so i want to find f prime of x is going to be equal to 1 over 2 let's open the brackets let's make it a big one so this whole thing is raised the power half so in differentiating what do i have i have i can bring my 1 over 2 i still have my x then this is going to be what's half minus 1 so i'm going to get x squared plus 1 to the power minus 1 over 2 plus then okay so not plus then don't forget we have to differentiate this in the bracket the x squared so then that will give me what 2x so that gives me times 2x times 2x here plus let's differentiate this part as well and this part as well is going to give us square root of what s squared plus one plus you have lane so when you are differentiating lane this whole thing is going to give us what one over x x plus don't forget you have lane updates so the differential of this is going to give us one over x plus square root of words x squared plus 1 then we multiply by 1 plus 1 over 2 x squared plus 1 divided by words to the power 1 on 2 this is going to give us what this then minus 1 is going to give us that times 2 x as well then so this is what we have.

04:16 Now let's simplify you realize that with this this can cancel that so that i could still have 1 over 2 here.

04:31 Let's open our brackets.

04:33 I have this squared times 0 squared which is going to give us what x squared so i have s squared.

04:41 This is going to because it is negative we are going to to get square root square root of what x squared plus one for that part plus our square root of what s squared plus one here plus this part as well so if you simplify this part this part don't forget we are going to get one divided by x plus plus one my square root of what x squared plus one.

05:25 Open brackets.

05:27 Simplified this and this is going to give me, don't forget this and that's cancel so i also have 1 plus x divided by what square root of what x plus 1...

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