Question

Can someone help me with this integral?

Answer 1

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

Answer 2

that e is to the power of \[\pi x\]

Answer 3

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?

Answer 4

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

Answer 5

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

Answer 6

just plug in your limits and your done :)

Answer 7

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

Answer 8

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

Answer 9

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

\[\frac{ -1 }{ \pi }e ^{-2 \pi}-\frac{ -1 }{ \pi }e ^{-2(0)} = \frac{ -1 }{ \pi } \left( e ^{-2 \pi } - 1\right)\]
\[=\frac{ 1 }{ \pi } \left( 1 -e ^{-2 \pi } \right) \]

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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 = )

Answer 14

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 :)

Answer 15

sorry one moment trying to make sense of all this lol

Answer 16

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?

Answer 17

@Tolio

Answer 18

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

Answer 19

http://tutorial.math.lamar.edu/Classes/CalcI/ComputingIndefiniteIntegrals.aspx

Answer 20

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

Answer 21

you're quite welcome :)

Answer 22

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Answer 23

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Answer 24

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Answer 25

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Answer 26

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Answer 27

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Answer 28

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Answer 30

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Answer 31

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Answer 32

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Answer 34

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They could not agree on a definition.

Answer 35

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Answer 36

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Answer 37

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Answer 39

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Answer 40

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Answer 41

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