The integral $\int x \left( \frac { 1 } { 2 } \right) ^ { x } d x$ is

A) $\frac { 2 ^ { x } ( x \ln 2 - 1 ) } { ( \ln 2 ) ^ { 2 } } + C$
B) $- \frac { 2 ^ { x } ( x \ln 2 + 1 ) } { ( \ln 2 ) ^ { 2 } } + C$
C) $- \frac { \left( \frac { 1 } { 2 } \right) ^ { x } ( x \ln 2 - 1 ) } { ( \ln 2 ) ^ { 2 } } + C$
D) $- \frac { \left( \frac { 1 } { 2 } \right) ^ { x } ( x \ln 2 + 1 ) } { ( \ln 2 ) ^ { 2 } } + C$
E) $\frac { \left( \frac { 1 } { 2 } \right) ^ { x } ( x \ln 2 + 1 ) } { ( \ln 2 ) ^ { 2 } } + C$

Question

The integral $\int x \left( \frac { 1 } { 2 } \right) ^ { x } d x$ is

A) $\frac { 2 ^ { x } ( x \ln 2 - 1 ) } { ( \ln 2 ) ^ { 2 } } + C$
B) $- \frac { 2 ^ { x } ( x \ln 2 + 1 ) } { ( \ln 2 ) ^ { 2 } } + C$
C) $- \frac { \left( \frac { 1 } { 2 } \right) ^ { x } ( x \ln 2 - 1 ) } { ( \ln 2 ) ^ { 2 } } + C$
D) $- \frac { \left( \frac { 1 } { 2 } \right) ^ { x } ( x \ln 2 + 1 ) } { ( \ln 2 ) ^ { 2 } } + C$
E) $\frac { \left( \frac { 1 } { 2 } \right) ^ { x } ( x \ln 2 + 1 ) } { ( \ln 2 ) ^ { 2 } } + C$

Quizplus · Accepted Answer

The integral can be solved using integration by parts. Let $ u = x $ and $ dv = \left( \frac{1}{2} \right)^x dx $. Then $ du = dx $ and $ v = -\frac{1}{\ln 2} \left( \frac{1}{2} \right)^x $. Applying integration by parts, the result is $- \frac { \left( \frac { 1 } { 2 } \right) ^ { x } ( x \ln 2 + 1 ) } { ( \ln 2 ) ^ { 2 } } + C$.

The Integral ∫x(12)xdx\int x \left( \frac { 1 } { 2 } \right) ^ { x } d x∫x(21​)xdx

The Integral $\int x \left( \frac { 1 } { 2 } \right) ^ { x } d x$