## bettyboop8904 2 years ago Can someone help me with this integral?

1. bettyboop8904

$\int\limits_{0}^{2} \frac{ dx }{ e ^{\pi x} }$

2. bettyboop8904

that e is to the power of $\pi x$

3. bettyboop8904

any idea? I think i have to separate the fraction so that 1 is in the numerator and then maybe use "u" substitution? to be able to take the anti-derivative?

4. Tolio

don't need to substitute; the integrand is just e^(-pi*x) so the antiderivative is $\frac{ -1 }{ \pi }e^{-\pi x}$

5. bettyboop8904

no that wouldn't work because when you find du it equals 0 = (

6. Tolio

7. bettyboop8904

it says the answer is $\frac{ 1 }{ \pi } (1-e ^{-2\pi})$

8. bettyboop8904

can you show me how to do it step by step? in laments terms lol

9. Tolio

right! take the expression i gave for the antideriv above and evaluate it at 2 and 0 , ie. plug in 2 then plug in 0 and take the difference $\frac{ -1 }{ \pi }e ^{-2 \pi}-\frac{ -1 }{ \pi }e ^{-2(0)} = \frac{ -1 }{ \pi } \left( e ^{-2 \pi } - 1\right)$ $=\frac{ 1 }{ \pi } \left( 1 -e ^{-2 \pi } \right)$

10. Tolio

ur just muliplying thru by -1 as last step to reverse the terms inside the parentheses

11. bettyboop8904

oh ok = ) can you show me step by step how you did the antiderivative. I seem to be having trouble = (

12. Tolio

whenever u hv e to the power of some constant * x then the anti-derivative is the same expoential function divided by the constant $\int\limits_{}^{}e ^{a x}= \frac{ 1 }{ a } e ^{a x }$

13. bettyboop8904

is that a rule? like what you learn from the book or class? I might just have skimmed over it in my notes and forgot = )

14. Tolio

definitely one of the standard integration rules but not really worth memorizing as u can see it makes intuitive sense: Dx(e^x) = e^x and Dx(e^ax) = ae^ax so when taking anti-deriv you hv to put in constant in denominator to get back original function. it's all about thinking in reverse :)

15. bettyboop8904

sorry one moment trying to make sense of all this lol

16. bettyboop8904

ok so i guess where I'm confused is that i thought the antiderivative formula was $\frac{ n ^{x+1} }{ x+1 }$ Why is it so different with e and ln from other functions?

17. bettyboop8904

@Tolio

18. Tolio

it is confusing; in your defn, n is a variable and x is a constant, x^2 or x^4, etc. in exponential fncs the n would be the constant and x the variable which requires different treatment the following whole webpage is great to explain the derivation but just to understand the difference a little more concretely scroll down to the very bottom of the page :) http://tutorial.math.lamar.edu/Classes/CalcI/DiffExpLogFcns.aspx

19. Tolio
20. bettyboop8904

thank you so much this page is helping me out a lot = ) @Tolio

21. Tolio

you're quite welcome :)