Evaluate the integral. $\int ( - 8 x + 4 ) ^ { - 2 } d x$
A) $\frac { \left( - 8 x ^ { 2 } + 4 \right) ^ { - 1 } } { 8 } + C$
B) $\frac { ( - 8 x + 4 ) ^ { - 1 } } { 8 } + C$
C) $\frac { ( - 8 x + 4 ) ^ { - 3 } } { 16 } + C$
D) $\frac { ( - 8 x + 4 ) ^ { - 3 } } { 8 } + C$
E) $\frac { ( - 8 x + 4 ) ^ { - 1 } } { 16 } + C$

Question

Evaluate the integral.  $\int ( - 8 x + 4 ) ^ { - 2 } d x$  
A)  $\frac { \left( - 8 x ^ { 2 } + 4 \right) ^ { - 1 } } { 8 } + C$
B)   $\frac { ( - 8 x + 4 ) ^ { - 1 } } { 8 } + C$
C)   $\frac { ( - 8 x + 4 ) ^ { - 3 } } { 16 } + C$
D)   $\frac { ( - 8 x + 4 ) ^ { - 3 } } { 8 } + C$
E)   $\frac { ( - 8 x + 4 ) ^ { - 1 } } { 16 } + C$

Quizplus · Accepted Answer

This integral can be solved by recognizing it as a form suitable for direct application of the power rule for integration, $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$. Here, $n = -2$, and the derivative of the inner function $-8x + 4$ is $-8$, which matches the adjustment needed for the chain rule. Thus, integrating $( - 8 x + 4 ) ^ { - 2 }$ gives $\frac{( - 8 x + 4 ) ^ { - 1 }}{8} + C$.

Evaluate the Integral ∫(−8x+4)−2dx\int ( - 8 x + 4 ) ^ { - 2 } d x∫(−8x+4)−2dx

Evaluate the Integral $\int ( - 8 x + 4 ) ^ { - 2 } d x$