Solve the problem.
-Evaluate
$$\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \frac { 1 } { \left( 1 + x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } } d x d y$$
A) $\frac { \pi } { 4 }$
B) $\frac { \pi } { 1 }$
C) $\frac { \pi } { 12 }$
D) $\frac { \pi } { 3 }$

Question

Quizplus · Accepted Answer

We can convert the given integral into polar coordinates as:
$$\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \frac { 1 } { \left( 1 + x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } } d x d y = \int_{0}^{\infty}\int_{0}^{\frac{\pi}{2}} \frac{r}{(1+r^2)^2}d\theta dr$$
Now, we can evaluate this integral as:
\begin{align*}
\int_{0}^{\infty}\int_{0}^{\frac{\pi}{2}} \frac{r}{(1+r^2)^2}d\theta dr &= \frac{\pi}{2}\int_{0}^{\infty} \frac{r}{(1+r^2)^2}dr \
&= \frac{\pi}{2}\int_{0}^{\infty} \frac{d(1+r^2)}{(1+r^2)^2} \
&= \frac{\pi}{2}\left[ -\frac{1}{1+r^2} \right]_{0}^{\infty} \
&= \frac{\pi}{2}\cdot1 \
&= \frac{\pi}{2}\cdot\frac{1}{2}\cdot2 \
&= \frac{\pi}{4}
\end{align*}
Therefore, the answer is (A).

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