Mathematics
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When using Integration by Parts, what is the best order of priority for choosing u and dv? Is ILATE foolproof?
In my recent mock exam, I had to evaluate ∫ e^(2x) * sin(3x) dx. When applying the ILATE rule, it led to a looping integral. Is there an easier systematic approach or tabular method to solve these cyclic integrals?
Recommended Answer by Prof. K. Venkatesh
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M.Sc. Mathematics, 18+ yrs IIT Faculty
ILATE is reliable, but for cyclic products like Exponential × Trigonometric:
• Set I = ∫ e^(2x) sin(3x) dx. Apply integration by parts twice, keeping the choice of u consistent (choose trigonometric as u both times).
• After round 2, I reappears on the RHS with a negative coefficient. Simply transpose I to LHS: I + (4/9)I = (13/9)I = ... and divide across. This is standard 6-mark board exam technique!
• Set I = ∫ e^(2x) sin(3x) dx. Apply integration by parts twice, keeping the choice of u consistent (choose trigonometric as u both times).
• After round 2, I reappears on the RHS with a negative coefficient. Simply transpose I to LHS: I + (4/9)I = (13/9)I = ... and divide across. This is standard 6-mark board exam technique!
Community Answers & Discussion (2)
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PKVerified Tutor • 46 mins agoILATE is reliable, but for cyclic products like Exponential × Trigonometric:
• Set I = ∫ e^(2x) sin(3x) dx. Apply integration by parts twice, keeping the choice of u consistent (choose trigonometric as u both times).
• After round 2, I reappears on the RHS with a negative coefficient. Simply transpose I to LHS: I + (4/9)I = (13/9)I = ... and divide across. This is standard 6-mark board exam technique! -
ASStudent • 46 mins agoThe D-I tabular method also works wonders here! You stop differentiating when the product of the row matches a multiple of the original integrand.
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