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How to calculate the integral ∫[(x^2)e^(5x)]dx?
Using integration by parts(or tabular method).
3 Answers
- MathPhDLv 67 years ago
Tabular method gives an easy formula that leads to the answer. In general, the integral of x^n*e^(mx) will be some polynomial of degree n times e^(mx) (then plus C). So here, the answer is (A1*x^2+A2*x+A3)e^(5x)+C and taking derivatives gives
x^2*e^(5x) = ((2A1*x+A2)e^(5x)+5(A1*x^2+A2*x+A3)e^(5x).
Simplifying x^2=5A1*x^2+(5A2+2A1)x+(5A3+A2).
Yipes! 3 equations and 3 unknowns! But it is easy! Find A1 first, then A2 from it, then A3.
- 7 years ago
Note: Take x^2 as the first function while integrating by parts to avoid the increase of its power.
