Correct Answer is : x6x^6x6 [(log⁡x)36[( x)³/6[6(logx)3 - 112(log⁡x)2+136log⁡x1/12 ( x)² + 1/36 x121(logx)2+361logx - 1216]1/216]2161] + c

x^6 [( x)³/12 - 1/6 ( x)² + 1/6 x - 1/36] + c

x^6 [( x)³/6 - 1/18 ( x)² + 1/12 x - 1/36] + c

x^6 [( x)³/6 - 1/12 ( x)² + 1/12 x + 1/36] + c

x^6 [( x)³/6 - 1/12 ( x)² + 1/36 x - 1/216] + c

Correct Answer is : x6x^6x6 [(log⁡x)36[( x)³/6[6(logx)3 - 112(log⁡x)2+136log⁡x1/12 ( x)² + 1/36 x121(logx)2+361logx - 1216]1/216]2161] + c

Test Index

AP EAMCET Engineering 2018 Apr 24 Shift 1 Paper

Section: Mathematics

Question : 75 of 160

Marks: +1, -0

∫

(\log x)^3 x^5

dx =

$x^6$ $[\frac{(\log x)^3}{12}$ - $\frac{1}{6} (\log x)^2 + \frac{1}{6} \log x$ - $\frac{1}{36}]$ + c
$x^6$ $[\frac{(\log x)^3}{6}$ - $\frac{1}{18} (\log x)^2 + \frac{1}{12} \log x$ - $\frac{1}{36}]$ + c
$x^6$ $[\frac{(\log x)^3}{6}$ - $\frac{1}{12} (\log x)^2 + \frac{1}{12} \log x$ + $\frac{1}{36}]$ + c
$x^6$ $[\frac{(\log x)^3}{6}$ - $\frac{1}{12} (\log x)^2 + \frac{1}{36} \log x$ - $\frac{1}{216}]$ + c

Solution:

\int (\log x)^3 x^5 dx

Use integration by parts

= (\log x)^3 \int x^5 dx - \int \left[ \frac{d}{dx} (\log x)^3 \int x^5 dx \right] dx

= (\log x)^3 \frac{x^6}{6} - \int \frac{3(\log x)^2}{x} \frac{x^6}{6} dx

= \frac{x^6 (\log x)^3}{6} - \frac{1}{2} \left[ (\log x)^2 \int x^5 dx \right]

- \int \left\{ \frac{d}{dx} (\log x)^2 \int x^5 dx \right\} dx

= \frac{x^6 (\log x)^3}{6} - \frac{x^6}{12} (\log x)^2

+ \frac{1}{6} \int (\log x) x^5 dx

Solve further,

= \frac{x^6 (\log x)^3}{6} - \frac{x^6}{12} (\log x)^2 + \frac{1}{6}

[\log x \int x^5 dx

- \int \left\{ \frac{d}{dx} (\log x) \int x^5 dx \right\} dx]

= \frac{x^6 (\log x)^3}{6} - \frac{x^6}{12} (\log x)^2 + \frac{x^6}{36} (\log x)

- \frac{1}{36} \int \left( \frac{x^6}{6} \right) + C

= x^6 \left[ \frac{(\log x)^3}{6} - \frac{(\log x)^2}{12} + \frac{\log x}{36} - \frac{1}{216} \right] + C

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