#8747; #160;x+1ex #160;.log #160;x #160;ex #160;dxLet #160;x #160;ex=t #8658;x.ex+1.exdx=dt #8756; #8747; #160;x+1ex #160;.log #160 ...

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Log Function

∫ x +1 ex log...

Question

$\int (x + 1) e^{x} \log (x e^{x}) d x$

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Solution

$\int (x + 1) e^{x} . \log (x e^{x}) d x Let x e^{x} = t \Rightarrow (x . e^{x} + 1 . e^{x}) d x = d t ∴ \int (x + 1) e^{x} . \log (x e^{x}) d x = \int \underset{II}{1} . \underset{I}{\log (t)} d t = \log t \int 1 d t - \int \{\frac{d}{d t} (\log t) - \int 1 d t\} d t = \log (t) \times t - \int \frac{1}{t} \times t d t = t \log (t) - t + C . . . (1) Substituting the value of t in eq (1) \Rightarrow \int (x + 1) e^{x} . \log (x e^{x}) d x = (x e^{x}) . \log (x e^{x}) - x e^{x} + C = x e^{x} \{\log (x e^{x}) - 1\} + C$

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