Evaluate the integral.
-$$\int ( 2 x - 1 ) \ln ( 14 x ) d x$$
A) $\left( x ^ { 2 } - x \right) \ln 14 x - x ^ { 2 } + x + C$
B) $\left( x ^ { 2 } - x \right) \ln 14 x - \frac { x ^ { 2 } } { 2 } + x + C$
C) $\left( x ^ { 2 } - x \right) \ln 14 x - \frac { x ^ { 2 } } { 2 } + 2 x + C$
D) $\left( \frac { x ^ { 2 } } { 2 } - x \right) \ln 14 x - \frac { x ^ { 2 } } { 4 } + x + C$

Question

Quizplus · Accepted Answer

This integral can be solved using integration by parts, where $u = \ln(14x)$ and $dv = (2x - 1)dx$. After integrating $dv$ to get $v = x^2 - x$ and differentiating $u$ to get $du = \frac{1}{x}dx$, applying the integration by parts formula $\int u dv = uv - \int v du$ gives $(x^2 - x)\ln(14x) - \int(x^2 - x)\frac{1}{x}dx$. Simplifying the integral part gives $-\frac{x^2}{2} + x$, leading to the final answer of $(x^2 - x)\ln(14x) - \frac{x^2}{2} + x + C$.

Evaluate the Integral ∫(2x−1)ln⁡(14x)dx\int ( 2 x - 1 ) \ln ( 14 x ) d x∫(2x−1)ln(14x)dx

Evaluate the Integral $\int ( 2 x - 1 ) \ln ( 14 x ) d x$