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Find \(\int_2^3 \,2x^2 \,e^{x^3} \,dx\).(a) \(e^{27}-e^8\)(b) \(\frac{2}{3} (e^{27}-e^8)\)(c) \(\frac{2}{3} (e^8-e^{27})\)(d) \(\frac{2}{3} (e^{27}+e^8)\)This question was addressed to me during a job interview.I would like to ask this question from Evaluation of Definite Integrals by Substitution topic in section Integrals of Mathematics – Class 12 |
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Answer» CORRECT choice is (B) \(\frac{2}{3} (e^{27}-e^8)\) EXPLANATION: I=\(\int_2^3 \,2x^2 \,e^{x^3} \,dx\) LET x^3=t Differentiating w.r.t x, we get 3x^2 dx=dt x^2 dx=\(\frac{dt}{3}\) The new limits When x=2, t=8 When x=3, t=27 ∴\(\int_2^3 \,2x^2 \,e^{x^3} \,dx=\frac{2}{3} \int_8^{27} \,e^t \,dt\) =\(\frac{2}{3} [e^t]_8^{27}=\frac{2}{3} (e^{27}-e^8).\) |
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