Define for . Then…

MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation

Problem

Define P(x)=x2∫0xln⁡(1+t) dt P(x)=x^{2}\displaystyle\int_{0}^{x} \ln(1+t)\,dt\, for x>−1x> -1. Then P′′(x)P''(x) equals (a) 2∫0x ⁣ln⁡(1+t) dt  +  2xln⁡(1+x)  +  x21+x2\displaystyle\int_{0}^{x}\!\ln(1+t)\,dt\; +\; 2x\ln(1+x)\; +\; \dfrac{x^{2}}{1+x} (b) 2∫0x ⁣ln⁡(1+t) dt  +  4xln⁡(1+x)2\displaystyle\int_{0}^{x}\!\ln(1+t)\,dt\; +\; 4x\ln(1+x) (c) 2∫0x ⁣ln⁡(1+t) dt  +  4xln⁡(1+x)  +  x21+x2\displaystyle\int_{0}^{x}\!\ln(1+t)\,dt\; +\; 4x\ln(1+x)\; +\; \dfrac{x^{2}}{1+x} (d) 2∫0x ⁣ln⁡(1+t) dt  +  4xln⁡(1+x)  +  x2\displaystyle\int_{0}^{x}\!\ln(1+t)\,dt\; +\; 4x\ln(1+x)\; +\; x

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