Evaluate the improper integral or state that it is divergent.
-$$\int _ { - \infty } ^ { 0 } \frac { d x } { ( 25 + x ) \sqrt { x } }$$
A) $\frac { \pi } { 5 }$
B) 0
C) $- \frac { \pi } { 5 }$
D) $- 5 \pi$

Question

Quizplus · Accepted Answer

We start with a substitution: Let $u = \sqrt{x}$ so that $du = \frac{1}{2\sqrt{x}}dx$ and $x = u^2.$ Also, as $x \rightarrow -\infty, u \rightarrow 0.$ Hence, 
$$\begin{aligned} \int_{-\infty}^{0}\frac{dx}{(25+x)\sqrt{x}} &= \int_{\infty}^{0}\frac{2u\,du}{(25+u^2)u} \ &= -2\int_{0}^{\infty}\frac{du}{25+u^2} \ &= -\frac{2}{5}\left[\arctan\left(\frac{u}{5}\right)\right]_{0}^{\infty} \ &= -\frac{2}{5}\left[\frac{\pi}{2} - \arctan(0)\right] \ &= -\frac{\pi}{5}. \end{aligned}$$

Evaluate the Improper Integral or State That It Is Divergent ∫−∞0dx(25+x)x\int _ { - \infty } ^ { 0 } \frac { d x } { ( 25 + x ) \sqrt { x } }∫−∞0​(25+x)x​dx​

Evaluate the Improper Integral or State That It Is Divergent $\int _ { - \infty } ^ { 0 } \frac { d x } { ( 25 + x ) \sqrt { x } }$