Question

$\int_{}^{}x^{3}$logx dx =

• A. $\frac{x^{4}\log x}{4}$ + c
• B. $\frac{1}{16}$ [4x⁴logx– x⁴] + c
• C. $\frac{1}{8}$ [x⁴ logx – 4x²] +c
• D. $\frac{1}{16}$ [4x⁴ logx + x⁴] + c

Answer 1

Answer: (B)

$\frac{1}{16}$ $4x⁴logx– x⁴$ + c

Answer 2

$\int_{}^{}x^{3}$logxdx=logx$\int_{}^{}x^{3}$dx$\int_{}^{}{\left\{ \frac{d}{dx}(\log x).\int_{}^{}{x^{3}dx} \right\} dx}$

=$\frac{x^{4}}{4}$logx – $\int_{}^{}\frac{1}{x}.\frac{x^{4}}{4}dx$ = $\frac{x^{4}}{4}$log x – $\frac { 1 } { 4 }$ $\int_{}^{}x^{3}$dx

= x⁴ + c ̃$\frac{1}{16}$ [4x⁴ log x – x⁴] + c