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)=ex‌log‌x Using the chain rule and product rule, we differentiate:
‌
d
dx
xx=‌
d
dx
(ex‌log‌x)=ex‌log‌x(x‌
1
x
+log‌x)=ex‌log‌x(1+log‌x)=xx(1+log‌x).
This computation shows that the derivative of xx is indeed xx(1+log‌x) : ‌
d
dx
xx=xx(1+log‌x)‌. ‌ Now, let's integrate both sides: ∫‌
d
dx
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