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JEE Main 2014 · 11 Apr · Shift 1, Mathematics question 76

If for n≥1n\ge1, Pn=∫1e(log⁡x)n dxP_n=\displaystyle\int_1^e(\log x)^n\,dx, then P10−90P8P_{10}-90P_8 is equal to :

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Answer and worked solution

Official answer: C

By parts, Pn=e−nPn−1P_n=e-nP_{n-1}. So P10=e−10P9=e−10(e−9P8)=−9e+90P8P_{10}=e-10P_9=e-10(e-9P_8)=-9e+90P_8, and
P10−90P8=−9e.P_{10}-90P_8=-9e.

Topics:Properties of definite integralsIntegration by parts
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