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jamesd2008
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Could anyone explain to me the rule when Integrating of exponentials with a fractional power?
e.g. ue to the power of -t/T. ue^-t/T
Thanks in advance
e.g. ue to the power of -t/T. ue^-t/T
Thanks in advance
Welcome to PF,jamesd2008 said:Could anyone explain to me the rule when Integrating of exponentials with a fractional power?
e.g. ue to the power of -t/T. ue^-t/T
Thanks in advance
The general formula for integrating exponentials with a fractional power is given by ∫ (x^a)e^x dx = (x^a)e^x - a∫ (x^(a-1))e^x dx, where a is any real number.
To handle the fractional power when integrating exponentials, we use the substitution method. We let u = x^a, then du = ax^(a-1) dx. We then substitute du and u into the integral, making it easier to solve.
Yes, there is a shortcut or trick for integrating exponentials with a fractional power. It is called the "integration by parts" method. This involves breaking down the integral into two parts and using a specific formula to solve for each part, making the overall integration easier.
Sure, an example of integrating an exponential with a fractional power is ∫ (x^(1/2))e^x dx. Using the substitution method, we let u = x^(1/2), then du = (1/2)x^(-1/2) dx. Substituting these into the integral, we get ∫ (x^(1/2))e^x dx = 2ue^x - ∫ e^x du = 2x^(1/2)e^x - 2∫ x^(-1/2)e^x dx. From here, we can use the integration by parts method to solve for the remaining integral.
Yes, there are special cases to consider when integrating exponentials with a fractional power. One important case is when the fractional power is equal to -1. In this case, the integral becomes ∫ e^x/x dx, which cannot be solved using the methods mentioned before. This is known as the "exponential integral" and has its own set of rules and techniques for solving it.