MHB How can substitution make solving integrals easier?

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Substitution, particularly u-substitution, simplifies the process of solving integrals by transforming complex expressions into more manageable forms. By letting u equal a function of x, such as u = e^(5x)/5, the integral can be restructured to make integration easier. This method also involves calculating the differential du, which helps in converting dx to du, further simplifying the integral. Understanding how to choose the right substitution is crucial for effective problem-solving in calculus. Mastering u-substitution can significantly enhance one's ability to tackle integral problems.
TheFallen018
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Hi, I've got this problem that I've been trying to work out. I think most of my problems come from the fact that I am not yet well versed in u substitution when it comes to integrals. I'm also not 100% sure what the problem is asking.

I've tried doing a couple of things, but they don't seem to be correct. I'm now at that point where everything I do confuses me more. If someone could help shed some light on the subject, I would be very grateful. Thanks.

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TheFallen018 said:
Hi, I've got this problem that I've been trying to work out. I think most of my problems come from the fact that I am not yet well versed in u substitution when it comes to integrals. I'm also not 100% sure what the problem is asking.

I've tried doing a couple of things, but they don't seem to be correct. I'm now at that point where everything I do confuses me more. If someone could help shed some light on the subject, I would be very grateful. Thanks.
A couple of things to get you started:

In the suggested substitution, if $u = \dfrac{e^{5x}}5$ then $u^2 = \dfrac{e^{10x}}{25}.$ That should help you with the denominator of the integrand.

If $u = \dfrac{e^{5x}}5$ then $\dfrac{du}{dx} = e^{5x}.$ That should help convert the $dx$ in the integral to something involving $du$.
 

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