# 2.6.62 inverse integrals with substitution

• MHB
• karush
In summary, the conversation is about a document on Overleaf that contains many custom macros which cannot be pasted into the code easily. The person is unsure about all the details but appreciates any comments. The topic of discussion is about the usage of b, x, and u in the document and there may be some errors in substitution. The conversation also mentions a mistake in the solution for problem 6.6.62 that should be corrected.
karush
Gold Member
MHB
View attachment 9266
ok this is from my overleaf doc
so too many custorm macros to just paste in code
but I think its ok,,, not sure about all details.

I got ? somewhat on b and x and u being used in the right places

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Last edited:
karush said:
ok this is from my overleaf doc
so too many custorm macros to just paste in code
but I think its ok,,, not sure about all details.

I got ? somewhat on b and x and u being used in the right places
Ummm...

-Dan

OK this one was kinda obvious but usually that is where I make the errors especially where there is substitution.
I post another one here soon.

karush said:
That all looks correct to me, except that at one point in the solution of 6.6.62 you have written $du=dx$. That should be $\dfrac{du}4 = dx$ (which is what you have correctly used in the following line).

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## 1. What is the concept of inverse integrals with substitution?

The concept of inverse integrals with substitution is a technique used in calculus to evaluate integrals that cannot be solved using basic integration methods. It involves substituting a variable with a new expression in order to simplify the integral and make it solvable.

## 2. How do you know when to use inverse integrals with substitution?

Inverse integrals with substitution are typically used when the integrand (the function being integrated) contains a complicated expression or a variable raised to a power. This technique can also be used when the integrand involves trigonometric functions or logarithmic functions.

## 3. What is the process for solving a 2.6.62 inverse integral with substitution?

The process for solving a 2.6.62 inverse integral with substitution involves first identifying the variable to be substituted, then selecting a suitable substitution expression. The next step is to rewrite the integral in terms of the new variable, and then solve the integral using basic integration techniques. Finally, the solution is converted back to the original variable.

## 4. What are some common mistakes to avoid when using inverse integrals with substitution?

One common mistake when using inverse integrals with substitution is incorrectly identifying the variable to be substituted. Another mistake is choosing an unsuitable substitution expression, which can result in a more complicated integral. It is also important to carefully convert the solution back to the original variable to avoid errors.

## 5. Can inverse integrals with substitution be used for definite integrals?

Yes, inverse integrals with substitution can be used for definite integrals. In this case, the limits of integration must also be converted to match the new variable. The solution will then be in terms of the new variable, so it must be converted back to the original variable before evaluating the definite integral.

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