Understanding 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

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Not sure how do I start from here, but do I let $$u = lnx$$ and substitute?
Cheers

 

[member 587159]

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?

 

[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?

 

[PeroK]

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.

 

[member 587159]

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.

 

[PeroK]

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

Are you asking if $25/5 = 5$?

 

[member 587159]

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

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]

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