To solve the integral
∫xx(1+log‌x)‌dx, we begin by recognizing that direct integration strategies such as substitution or integration by parts do not straightforwardly apply. Instead, we'll look for a pattern or simplification.
First, consider the term
xx. One way to differentiate
xx is using the exponential and logarithmic transformation. Recall that:
xx=elog(xx)=ex‌log‌xUsing the chain rule and product rule, we differentiate:
This computation shows that the derivative of
xx is indeed
xx(1+log‌x) :
‌xx=xx(1+log‌x)‌. ‌Now, let's integrate both sides:
∫‌xx‌dx=∫xx(1+log‌x)‌dx.This simplifies to:
xx=∫xx(1+log‌x)‌dx Therefore, the integral
∫xx(1+log‌x)‌dx evaluates to
xx+C, where
C is the constant of integration.
Comparing this to the options given, Option B is the correct answer:
xx+c