If ∫x4(log⁡x)3dx=x5[A(log⁡x)3+B(log⁡x)2+Clog⁡x+D]+k, thenA+B+C+5D=

=∫x4(log⁡x)3dxApply bypart will get=(log⁡x)3⋅∫x4⋅dx−∫ddx⋅(log⁡x)3⋅∫x4⋅dx⋅dx=(log⁡x)3⋅x55−∫3⋅log⁡x2⋅1x⋅x55⋅dx=(log⁡x)3⋅x55−∫3⋅log⁡x2⋅1x⋅x45⋅dx=(log⁡x)3⋅x55−25log⁡x2⋅∫x4⋅dx−∫ddxlog⁡x2⋅∫x4⋅dx⋅dx=(log⁡x)3⋅x55−25log⁡x2⋅x55−∫2log⁡x1x⋅x55⋅dx=(log⁡x)3⋅x55−25log⁡x2⋅x55−∫2log⁡x1x⋅x55⋅dx =(log⁡x)3⋅x55−2x525log⁡x2−425∫log⁡x⋅x4dx =(log⁡x)3⋅x55−2x525log⁡x2−425log⁡x⋅x55−∫1x⋅x55⋅dx=x5[15(log⁡x)3-225(log⁡x)2-4125log⁡x+4625]+k=225

If ∫x4(log⁡x)3dx=x5[A(log⁡x)3+B(log⁡x)2+Clog⁡x+D]+k, thenA+B+C+5D=

=∫x4(log⁡x)3dxApply bypart will get=(log⁡x)3⋅∫x4⋅dx−∫ddx⋅(log⁡x)3⋅∫x4⋅dx⋅dx=(log⁡x)3⋅x55−∫3⋅log⁡x2⋅1x⋅x55⋅dx=(log⁡x)3⋅x55−∫3⋅log⁡x2⋅1x⋅x45⋅dx=(log⁡x)3⋅x55−25log⁡x2⋅∫x4⋅dx−∫ddxlog⁡x2⋅∫x4⋅dx⋅dx=(log⁡x)3⋅x55−25log⁡x2⋅x55−∫2log⁡x1x⋅x55⋅dx=(log⁡x)3⋅x55−25log⁡x2⋅x55−∫2log⁡x1x⋅x55⋅dx =(log⁡x)3⋅x55−2x525log⁡x2−425∫log⁡x⋅x4dx =(log⁡x)3⋅x55−2x525log⁡x2−425log⁡x⋅x55−∫1x⋅x55⋅dx=x5[15(log⁡x)3-225(log⁡x)2-4125log⁡x+4625]+k=225

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If $\int x^{4} (\log x)^{3} d x = x^{5} [A (\log x)^{3} + B (\log x)^{2} + C \log x + D] + k, then A + B + C + 5 D =$

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$\frac{2}{25}$

$\frac{8}{25}$

$\frac{12}{25}$

$\frac{16}{25}$

answer is A.

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Detailed Solution

$= \int x^{4} (\log x)^{3} d x$

Apply bypart will get

$\begin{aligned} = (\log x)^{3} \cdot \int x^{4} \cdot d x - \int (\frac{d}{d x} \cdot (\log x)^{3} \cdot \int x^{4} \cdot d x) \cdot d x \\ = (\log x)^{3} \cdot \frac{x^{5}}{5} - \int 3 \cdot {(\log x)}^{2} \cdot \frac{1}{x} \cdot \frac{x^{5}}{5} \cdot d x \end{aligned}$

$\begin{aligned} = (\log x) 3 \cdot \frac{x^{5}}{5} - \int 3 \cdot {(\log x)}^{2} \cdot \frac{1}{x} \cdot \frac{x^{4}}{5} \cdot d x \\ = (\log x)^{3} \cdot \frac{x^{5}}{5} - \frac{2}{5} [{(\log x)}^{2} \cdot \int x^{4} \cdot d x - \int (\frac{d}{d x} {(\log x)}^{2} \cdot \int x^{4} \cdot d x) \cdot d x] \\ = (\log x)^{3} \cdot \frac{x^{5}}{5} - \frac{2}{5} [{(\log x)}^{2} \cdot \frac{x^{5}}{5} - \int 2 (\log x) \frac{1}{x} \cdot \frac{x^{5}}{5} \cdot d x] \end{aligned}$

$\begin{aligned} = (\log x)^{3} \cdot \frac{x^{5}}{5} - \frac{2}{5} [{(\log x)}^{2} \cdot \frac{x^{5}}{5} - \int 2 (\log x) \frac{1}{x} \cdot \frac{x^{5}}{5} \cdot d x] \end{aligned} \begin{aligned} = (\log x)^{3} \cdot \frac{x^{5}}{5} - \frac{2 x^{5}}{25} {(\log x)}^{2} - \frac{4}{25} \int (\log x) \cdot x^{4} d x \end{aligned} \begin{aligned} = (\log x)^{3} \cdot \frac{x^{5}}{5} - \frac{2 x^{5}}{25} {(\log x)}^{2} - \frac{4}{25} \end{aligned} [(\log x) \cdot \frac{x^{5}}{5} - \int \frac{1}{x} \cdot \frac{x^{5}}{5} \cdot d x]$