Question

Evaluate the integral
${\int }_1^e\frac{(\ln x{)}^3}{x}dx$

• A. $e^4/4$
• B. $\frac{1}{4}$
• C. $\frac{1}{4}(e^4-1)$
• D. $e^4-1$

Answer 1

Answer: (B)

$\frac{1}{4}$

${\int }_1^e\frac{(lnx{)}^3}{x}dx$

${\int }_1^e\left(lnx{)}^{3}d\right(lnx)=\frac{(lnx{)}^4}{4}+c$

${\int }_1^e\left(lnx{)}^{3}d\right(lnx)=(\frac{(1{)}^4}{4}+c)-(\frac{0}{4}+c)$

$=\frac{1}{4}$

Thus ${\int }_1^e\frac{(lnx{)}^3}{x}dx=\frac{1}{4}$