Let $a$ and $b$ be real numbers. What integral must appear in place of the question mark "?" to make the following statement true?
$$\int _ { - \infty } ^ { a } \frac { 10 } { x ^ { 2 } + 9 } d x + \int _ { a } ^ { \infty } \frac { 10 } { x ^ { 2 } + 9 } d x = ? + \int _ { b } ^ { \infty } \frac { 10 } { x ^ { 2 } + 9 } d x$$
A) $\int _ { - \infty } ^ { 2 } \frac { 9 } { x ^ { 2 } + 10 } d x$
B) $\int _ { b } ^ { - \infty } \frac { 10 } { x ^ { 2 } - 9 } d x$
C) $\int _ { - \infty } ^ { b } \frac { 10 } { x ^ { 2 } + 9 } d x$
D) $\int _ { b } ^ { - \infty } \frac { 10 } { x ^ { 2 } + 9 } d x$
E) none of these

Question

Quizplus · Accepted Answer

The two integrals on the left side of the equation add up to the integral of the function over the entire real line, since they cover the intervals from $-\infty$ to $a$ and from $a$ to $\infty$, respectively. Therefore, to balance the equation, the missing integral must cover the interval from $-\infty$ to $b$, with the same integrand $\frac{10}{x^2 + 9}$, making option C correct.

Let aaa And bbb Be Real Numbers A) ∫−∞29x2+10dx\int _ { - \infty } ^ { 2 } \frac { 9 } { x ^ { 2 } + 10 } d x∫−∞2​x2+109​dx

Let $a$ And $b$ Be Real Numbers A) $\int _ { - \infty } ^ { 2 } \frac { 9 } { x ^ { 2 } + 10 } d x$