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## TomLikesPhysics 2 years ago I need help with an Integral (Substitution).

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1. TomLikesPhysics

$\int\limits_{0}^{100}e^\sqrt{x}dx=?$ I am supposed to solve this using the method of substitution. I have no clue how the hell this would help in that weird case.

2. hartnn

u know integration by parts ?

3. across

Let$u=\sqrt x.\tag{1}$Then,$du=\frac1{2\sqrt x}dx\Longrightarrow dx=2\sqrt xdu=2udu.\tag{2}$It follows from $$(1)$$ that$x=0\Longrightarrow u=0\text{, and}\\ x=100\Longrightarrow u=10.$So, from $$(1)$$, $$(2)$$ and the new limits, we have that$2\int_0^{10}ue^u\,du.$

4. TomLikesPhysics

@hartnn Yes, that would be my first impulse but I should use the substitution method.

5. hartnn

as shown by @across

6. TomLikesPhysics

Ok, and after that I have to integrate by parts do integrate the $ue^udu$ part, right?

7. hartnn

yes.

8. across

Yes, or you could use the (not so) common knowledge that$\int xe^x\,dx=e^x(x-1).$

9. TomLikesPhysics

^^ I thought that I saw that Integral before, but not today it seems. Thx for your help.

10. hartnn

more general formula : $$\large \int e^x(f(x)+f'(x))dx=e^xf(x)+c$$

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