How can substitution make solving integrals easier?

In summary, the conversation discusses a problem with integrals and the speaker's lack of understanding in u substitution. They have tried various methods but are now confused and seeking help. Some suggestions are given for the substitution and its derivative to help with the integration.
  • #1
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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  • #2
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 $du$.
 

FAQ: How can substitution make solving integrals easier?

What is integration by substitution?

Integration by substitution is a technique used to evaluate integrals by substituting a variable with a new one. This allows for the integral to be rewritten in a simpler form, making it easier to solve.

When should integration by substitution be used?

Integration by substitution should be used when the integrand contains a complicated expression that can be simplified by substituting a variable. It is also useful for integrals where the integrand is a product of two functions, one of which is the derivative of the other.

What is the general formula for integration by substitution?

The general formula for integration by substitution is ∫f(g(x))g'(x) dx = ∫f(u) du, where u=g(x) and du=g'(x) dx.

What are the steps for integration by substitution?

The steps for integration by substitution are as follows: 1) Identify the function inside the integral that can be substituted with a new variable. 2) Choose a suitable substitution that will simplify the integral. 3) Rewrite the integral in terms of the new variable and its derivative. 4) Solve the new integral. 5) Substitute back the original variable to get the final answer.

What are some common mistakes to avoid when using integration by substitution?

Some common mistakes to avoid when using integration by substitution include: 1) Forgetting to substitute back the original variable in the final answer. 2) Choosing an incorrect substitution that does not simplify the integral. 3) Making a mistake in differentiating the new variable. 4) Not being aware of trigonometric identities when working with trigonometric functions. 5) Forgetting to include the constant of integration in the final answer.

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