Question at position 9 ∫033xex3dx=\int_0^33xe^{\frac{x}{3}}dx=903276

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Question at position 9 ∫033xex3dx=\int_0^33xe^{\frac{x}{3}}dx=903276

BlackTom AI · Accepted Answer

To begin, let's parse the integral and the limits involved. We are evaluating I = ∫ from 0 to 3 of x e^{x/3} dx. This requires integration by parts or a suitable substitution to handle the product of a polynomial and an exponential.
Option A: 9. This is the value obtained when carrying out the integration by parts correctly: choose u = x and dv = e^{x/3} dx, which leads to v = 3 e^{x/3} and du = dx. Then I = x·3 e^{x/3} |0^3 − ∫_0^3 3 e^{x/3} dx. The remaining integral equals 9(e^{1} − e^{0}) = 9(e − 1). Evaluating the boundary term gives 9e. Subtracting yields I = 9e − 9(e − 1) = 9. Therefore, 9 is the correct value.
Option B: 0. This would incorrectly imply the area under the curve cancels out over the interval, which is not plausible here since x e^{x/3} is nonnegative on [0,3]. The exponential factor is positive, and x is nonnegative, so the integral cannot be zero.
Option C: 3. This would suggest only a small portion of the total accumulation is counted. However, the integration by parts contributes terms that sum to 9 after simplification, so 3 misses the substantial contribution from both the boundary term and the exponential integral.
Option D: 27. Claiming 27 would indicate a much larger accumulation than the actual computation yields. The boundary and integral terms, when combined, cancel to leave 9, not 27.
Option E: 6. This value is halfway between 3 and 9 and could stem from a mistaken evaluation of the integral or a misapplication of the integration by parts steps. The precise calculation shows the result is 9, not 6.

Question at position 9 ∫033xex3dx=\int_0^33xe^{\frac{x}{3}}dx=903276

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