Understanding Integration by Substitution

In summary, we have discussed substituting ##u = 5\ln x## in order to solve the integral ##\int_1^{e^2} f(x) dx##, and then transforming the bounds of the integral to match the new variable. This allows us to simplify the integral to ##\frac{1}{5}\int_0^{10} f(u) du##. We also explored the idea of doing another substitution to solve for different bounds, but determined that it was not necessary in this case. Overall, this process makes the work easier for the original integral.
  • #1
jisbon
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Homework Statement
$$\int ^{10}_{0}f\left( x\right) dx=25$$
Find the value of
$$\int ^{e^{2}}_{1}\dfrac {f\left( 5\times \ln \left( x\right) \right) }{x}dx$$
Relevant Equations
-
Not sure how do I start from here, but do I let $$u = lnx$$ and substitute?
Cheers
 
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  • #2
Yes, let ##u = 5\ln x##. Then ##du = \frac{5}{x} dx \implies \frac{du}{5} = \frac{1}{x}dx## and your integral becomes

$$\frac{1}{5}\int_0^{10} f(u) du$$

Can you conclude?
 
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  • #3
Math_QED said:
Yes, let ##u = 5\ln x##. Then ##du = \frac{5}{x} dx \implies \frac{du}{5} = \frac{1}{x}dx## and your integral becomes

$$\frac{1}{5}\int_0^{10} f(u) du$$

Can you conclude?
$$\frac{1}{5}\int_0^{10} f(u) du$$

Since I'm trying to find from 1 to $$e^2$$ instead of 0 to 10, do I do another substitution?
 
  • #4
Can I say that $$1/5 *\int ^{10}_{0}f\left( x\right) dx=5$$ too?
 
  • #5
Math_QED said:
Yes, let ##u = 5\ln x##. Then ##du = \frac{5}{x} dx \implies \frac{du}{5} = \frac{1}{x}dx## and your integral becomes

$$\frac{1}{5}\int_0^{10} f(u) du$$

Can you conclude?
I think that is giving too much away. That is 90% of the work.
 
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  • #6
PeroK said:
I think that is giving too much away. That is 90% of the work.

Yes you may be right, the OP thought of the subsitution ##u = \ln x ## so at least he thought about the question a bit.
 
  • #7
jisbon said:
Can I say that $$1/5 *\int ^{10}_{0}f\left( x\right) dx=5$$ too?
Are you asking if ##25/5 = 5##?
 
  • #8
jisbon said:
$$\frac{1}{5}\int_0^{10} f(u) du$$

Since I'm trying to find from 1 to $$e^2$$ instead of 0 to 10, do I do another substitution?

Please reread my reply:

$$\int_1^{e^2} \dots dx = \int_{0}^{10}\dots du$$ after substitution ##u = 5 \ln x## (the bounds of the integral transform by the subsitution).
 
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  • #9
Sorry, I didn't understand why the bounds of the integral were transformed earlier on. All is good now, thanks all for your help
 

1. What is integration by substitution?

Integration by substitution is a technique used in calculus to evaluate integrals by substituting a variable with a different variable or function in order to make the integration process easier.

2. When should I use integration by substitution?

Integration by substitution is typically used when the integral contains a complicated function or a combination of functions that cannot be easily integrated using other methods. It can also be used to simplify integrals that involve trigonometric or exponential functions.

3. How do I perform integration by substitution?

The first step in integration by substitution is to identify a function to substitute, typically denoted as u, which can be easily differentiated. Then, substitute u and its derivative into the integral, simplify the integral using algebraic techniques, and finally integrate the resulting expression.

4. What are some common substitution strategies?

There are several common substitution strategies used in integration, including using a trigonometric substitution for integrals involving radicals, substitution of a polynomial for integrals involving polynomial expressions, and substitution of an exponential function for integrals involving exponential expressions.

5. Are there any tips for success with integration by substitution?

One tip for success with integration by substitution is to carefully choose the substitution variable, making sure it simplifies the integral rather than making it more complicated. Additionally, it is important to practice and become familiar with different substitution strategies in order to quickly identify the most efficient approach for a given integral.

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