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Exponential Functions
exdx=ex+C …(1) (e=2.71828...:the base of natural logarithm) ekxdx=ekx+C …(1’) axdx=+C (a>0, a≠1) …(2) Logarithmic Functions log x dx=x log x−x+C …(3) (↑ from integration by parts)
When the base is omitted in the logarithmic functions, the bases are understood as e. To denote the natural logarithm, the symbol ln x is often used. (natural logarithm)
(Because)(1)← (ex)=ex → exdx=ex+C (1’)← (ekx)=kekx→(ekx)=ekx →ekxdx=ekx+C (2)← As ax=exlog a, according to (1’) axdx=exlog adx=exlog a+C=+C (3)← (x log x−x)=log x+x −1=log x+1−1=log x → log xdx=x log x−x+C |
Examples 1.e3xdx=e3x+C ←(1’) 2.2xdx=+C ←(2) 3.log 3x dx=(log 3+log x)dx =x log 3+x log x−x+C ←(3) =x(log 3+log x)−x+C=x log 3x−x+C
In this page, only the basic integrals which don’t require neither the integration by substitution nor the integration by parts are treated.
Many of Integrals of Exponential and Logarithmic Functions appear at the integration by substitution or the integration by parts. |
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QuestionFind the integrals.
(Step 1)Select one of the questios from the left column.
(Step 2)Select the integral corresponding to the question. (Repeat) Step 1→Step 2 |
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