Question

$\int32x^{3}\left(log x\right)^{2}dx$ is equal to

• A. $ 8x^{4}\left(log x\right)^{2} + C $
• B. $ x^{4}\left\{8\left(log x\right)^{2} - 4log x + 1\right\} + C$
• C. $ x^{4}\left\{8\left(log x\right)^{2} - 4log x \right\} + C$
• D. $ x^{3}\left\{\left(log x\right)^{2} +2log x \right\} + C$

Answer 1

Answer: (B)

$\int32x^{3}\left(log x\right)^{2}dx $

$ 32\left\{\left(log x^{2}\right) \frac{x^{4}}{4}-\int2log x \cdot\frac{1}{x}\cdot\frac{x^{4}}{4}dx\right\}+C' $

$ =8x^{4}\left(logx\right)^{2}-16\int x^{3}log x dx+C' $

$= 8x^{4}\left(log x\right)^{2}-16\left\{logx \cdot\frac{x^{4}}{4}-\int\frac{1}{x}\cdot\frac{x^{4}}{4}dx \right\}+C'' $

$ = 8x^{4}\left(log x\right)^{2}-4x^{4}logx +4 \int x^{3}dx +C'' $

$ =8x^{4}\left(logx\right)^{2} +4x^{4}logx+x^{4}+C$

$ = x^{4 }\left[8\left(logx\right)^{2}-4logx +1\right]$