# Integration problem using substitution

• chwala
In summary: You're welcome.In summary, the conversation is about solving an indefinite integral using the substitution method with u = sin 4x. The question asks for the exact value of the integral and the conversation involves finding the appropriate substitution and limits, as well as splitting the integrand into separate terms. Eventually, the solution is found to be 0.115.
chwala
Gold Member

## Homework Statement

using ## u= sin 4x## find the exact value of ##∫ (cos^3 4x) dx##[/B]

## The Attempt at a Solution

## u= sin 4x## [/B]on integration ##u^2/2=-cos4x/4 ## , →##-2u^6={cos 4x}^3 ##...am i on the right track because now i end up with ##∫{{-2u^6}/{4.-2u^2}}du## or should i use

##du=4cos 4x dx ## to end up with ## 0.25 ∫ cos^24x du## which looks wrong to me

chwala said:

## u= sin 4x##
on integration ##u^2/2=-cos4x/4 ## ,
No, you've integrated one side wrt u and the other wrt x.

haruspex said:
No, you've integrated one side wrt u and the other wrt x.
so should i use the second approach?

chwala said:
so should i use the second approach?
Yes, but you need to get all of the references to x turned into references to u.

You are asked for an exact value, but it is an indefinite integral. Remember that the limits need to be expressed in terms of u as well.

yes the limits are from 0 to π/24

is ## 0.25∫{cos^24x}du## correct?

chwala said:
the limits are from 0 to π/24
So what are the limits on u?
chwala said:
is ## 0.25∫{cos^24x}du## correct?
Yes.

haruspex said:
So what are the limits on u?

Yes.
limits on u are 0 to 30, now how do i proceed with the integration?

## cos^2 4x## = ##(cos 8x+1)##/2

should we substitute again? or are we going to have##0.25∫cos^24x d {sin4x} ##

haruspex said:
So what are the limits on u?

Yes.
i am a bit confused we cannot integrate a variable say ##x## with respect to another variable say ##u##, i am stuck here

chwala said:
limits on u are 0 to 30,
No.
chwala said:
how do i proceed with the integration?
You have the cos2 of some angle, and you need to express that in terms of u, the sine of the same angle. Does nothing click?

haruspex said:
No.

You have the cos2 of some angle, and you need to express that in terms of u, the sine of the same angle. Does nothing click?
sorry limits are from 0 to 0.5 an oversight on my part...

haruspex said:
No.

You have the cos2 of some angle, and you need to express that in terms of u, the sine of the same angle. Does nothing click?
i now get it lol
## 0.25∫{1-u^2}du ## from u=0 to u=0.5 thanks mate solution is ## 0.115##

Why don't you try splitting ##cos^34x## into ##cos4x## and another term containing the term used for ##u## substitution?

Eclair_de_XII said:
Why don't you try splitting ##cos^34x## into ##cos4x## and another term containing the term used for ##u## substitution?
i have seen the obstacle with that move...

chwala said:
i have seen the obstacle with that move...
i have seen it, check post 13

Eclair_de_XII said:
Why don't you try splitting ##cos^34x## into ##cos4x## and another term containing the term used for ##u## substitution?
chwala said:
i have seen the obstacle with that move...
@chwala
Actually that is the move you finally made to solve. Check the time of @Eclair_de_XII 's post and your posts.

## 1. What is the general process for solving an integration problem using substitution?

The general process for solving an integration problem using substitution is as follows:

1. Identify the function within the integral that can be substituted.
2. Choose an appropriate substitution variable and set up the substitution equation.
3. Apply the substitution to the entire integral.
4. Simplify the resulting integral and solve for the variable.
5. Reverse the substitution and simplify the final solution.

## 2. How do you know when to use substitution in an integration problem?

Substitution is typically used when the integrand contains a function and its derivative, or when the integrand contains a function raised to a power. In these cases, substitution can help simplify the integral and make it easier to solve.

## 3. Can substitution be used for all integration problems?

No, substitution cannot be used for all integration problems. It is most effective when the integrand contains a function and its derivative, or when the integrand contains a function raised to a power. In other cases, other integration techniques may be more appropriate.

## 4. Are there any special substitution rules to keep in mind when solving an integration problem?

Yes, there are a few special substitution rules to keep in mind when solving an integration problem:

• If the integrand contains a square root, choose a substitution that will eliminate the square root.
• If the integrand contains a trigonometric function, choose a substitution that will convert it to a standard form.
• If the integrand contains a rational function, choose a substitution that will eliminate any fractions.

## 5. What are some common mistakes to avoid when using substitution to solve an integration problem?

Some common mistakes to avoid when using substitution to solve an integration problem include:

• Choosing the wrong substitution variable.
• Forgetting to include the differential, dx, in the substitution equation.
• Not simplifying the resulting integral before solving for the variable.
• Forgetting to reverse the substitution when simplifying the final solution.

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