# Integration by substitution?

jisbon
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

## Answers and Replies

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?

Delta2
jisbon
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?

jisbon
Can I say that $$1/5 *\int ^{10}_{0}f\left( x\right) dx=5$$ too?

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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.

Delta2
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.

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Can I say that $$1/5 *\int ^{10}_{0}f\left( x\right) dx=5$$ too?
Are you asking if ##25/5 = 5##?

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

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).

jisbon
jisbon
Sorry, I didn't understand why the bounds of the integral were transformed earlier on. All is good now, thanks all for your help