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

Question

Quizplus · Accepted Answer

First, integrate with respect to y:
$$\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { \infty } \frac { 1 } { x \left( y ^ { 2 } + 1 \right) } d y d x = \int_{1}^{4} \left[\frac{1}{x}\tan^{-1}(y)\right]_{0}^{\infty} dx = \int_{1}^{4} \frac{\pi}{2x} dx = \frac{\pi}{2} \ln 4$$
So the answer is C.

Evaluate the Improper Integral ∫14∫0∞1x(y2+1)dydx\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { \infty } \frac { 1 } { x \left( y ^ { 2 } + 1 \right) } d y d x∫14​∫0∞​x(y2+1)1​dydx

Evaluate the Improper Integral $\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { \infty } \frac { 1 } { x \left( y ^ { 2 } + 1 \right) } d y d x$