How Does U-Substitution Simplify Integration Problems?

Thread starter QuarkCharmer
Start date Jul 5, 2011
Tags

Integrals

Anyway, thanks for that tip. I'll be sure to keep it in mind. I just have to remember that the pattern is only in the form you showed, and not any other form. I'm sure I'll get to see how the form can be changed later, and it will make more sense. Just at this point, I'm not sure how to change the form, so I think that I'll just have to remember this form for now.In summary, when integrating expressions involving u-substitution, it is important to check if the integral can be rewritten in the form of \int [f(x)]^n f'(x)\, dx. This will determine if the substitution u=f(x) will work,

Jul 6, 2011

Ray Vickson

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QuarkCharmer said:

Related to this question:

[tex]\int sec^{3}(x)tan(x) dx[/tex]

[itex]u=sec(x)[/itex], and so, [itex]\frac{du}{dx}=sec(x)tan(x)[/itex], and [tex]dx=\frac{du}{sec(x)tan(x)}[/tex]

[tex]\int u^{3}tan(x) dx[/tex]
[tex]\int \frac{u^{3}tan(x)}{sec(x)tan(x)} du[/tex]
[tex]\int \frac{u^{3}}{sec(x)} du[/tex]

I have no idea what to do with this one? Can I put my u=sec back in and try again now?

Edit: Na, that just gets me back to where I started.

Oh wait, because u is equal to sec, can I just call that [itex]\frac{u^{3}}{u}[/itex] ?

You could note that [itex] f(x) = \sec^{3}(x) \tan(x)[/itex] has the form [itex] f(x) = \sin(x)/ \cos^{4}(x) [/itex], so substituting [itex] u = \cos(x) [/itex] gives [itex] f(x) dx = - du/u^4 [/itex].

RGV

<h2> What is u-substitution in integrals?</h2><p>U-substitution is a method used to simplify integrals by substituting a new variable, u, in place of an existing variable in the integrand. This allows for easier integration and can help solve more complex integrals.</p><h2> How do I know when to use u-substitution?</h2><p>U-substitution is most useful when the integrand contains a composite function, such as f(g(x)), where g(x) is the inner function. In these cases, u-substitution can be used to simplify the integrand and make integration easier.</p><h2> What is the general process for u-substitution?</h2><p>The general process for u-substitution involves three steps: 1) identify the inner function, g(x), in the integrand, 2) let u equal g(x) and find du/dx, and 3) substitute u and du into the integral, making sure to also change the limits of integration if necessary.</p><h2> Can u-substitution be used for definite integrals?</h2><p>Yes, u-substitution can be used for both indefinite and definite integrals. When using u-substitution for definite integrals, it is important to also change the limits of integration to match the new variable, u.</p><h2> Are there any common mistakes to avoid when using u-substitution?</h2><p>One common mistake when using u-substitution is forgetting to change the limits of integration to match the new variable, u. It is also important to take the derivative of u and substitute it in for du, rather than just using du/dx. Additionally, it is important to check the final answer by substituting the original variable, x, back in to ensure it matches the original integrand.</p>

FAQ: How Does U-Substitution Simplify Integration Problems?

What is u-substitution in integrals?

U-substitution is a method used to simplify integrals by substituting a new variable, u, in place of an existing variable in the integrand. This allows for easier integration and can help solve more complex integrals.

How do I know when to use u-substitution?

U-substitution is most useful when the integrand contains a composite function, such as f(g(x)), where g(x) is the inner function. In these cases, u-substitution can be used to simplify the integrand and make integration easier.

What is the general process for u-substitution?

The general process for u-substitution involves three steps: 1) identify the inner function, g(x), in the integrand, 2) let u equal g(x) and find du/dx, and 3) substitute u and du into the integral, making sure to also change the limits of integration if necessary.

Can u-substitution be used for definite integrals?

Yes, u-substitution can be used for both indefinite and definite integrals. When using u-substitution for definite integrals, it is important to also change the limits of integration to match the new variable, u.

Are there any common mistakes to avoid when using u-substitution?

One common mistake when using u-substitution is forgetting to change the limits of integration to match the new variable, u. It is also important to take the derivative of u and substitute it in for du, rather than just using du/dx. Additionally, it is important to check the final answer by substituting the original variable, x, back in to ensure it matches the original integrand.