Use a table of integrals to find the integral. $\int \frac { \sqrt { 49 - x ^ { 4 } } } { x } d x$
A)
$$\int \frac { \sqrt { 49 - x ^ { 4 } } } { x } d x = \frac { 1 } { 2 } \left[ \sqrt { 49 - x ^ { 4 } } - 7 \log \left( \sqrt { 49 - x ^ { 4 } } + 7 \right) + 49 \log \left( x ^ { 2 } \right) \right] + C$$
B)
$$\int \frac { \sqrt { 49 - x ^ { 4 } } } { x } d x = \frac { 1 } { 2 } \left[ \sqrt { 49 - x ^ { 4 } } - 7 \log \left( \sqrt { 49 - x ^ { 4 } } + 7 \right) + 7 \log \left( x ^ { 2 } \right) \right] + C$$
C)
$$\int \frac { \sqrt { 49 - x ^ { 4 } } } { x } d x = \frac { 1 } { 2 } \left[ \sqrt { 49 - x ^ { 4 } } - 49 \log \left( \sqrt { 49 - x ^ { 4 } } + 7 \right) + 7 \log \left( x ^ { 2 } \right) \right] + C$$
D)
$$\int \frac { \sqrt { 49 - x ^ { 4 } } } { x } d x = \frac { 1 } { 2 } \left[ \sqrt { 49 - x ^ { 4 } } - \log \left( \sqrt { 49 - x ^ { 4 } } + 1 \right) + 7 \log \left( x ^ { 2 } \right) \right] + C$$
E)
$$\int \frac { \sqrt { 49 - x ^ { 4 } } } { x } d x = \frac { 1 } { 2 } \left[ \sqrt { 49 - x ^ { 4 } } - 7 \log \left( \sqrt { 49 - x ^ { 4 } } + 1 \right) + 7 \log \left( x ^ { 2 } \right) \right] + C$$

Question

Quizplus · Accepted Answer

Using the substitution $x^2 = 7\sin	heta$ gives
\begin{align*}
\int \frac{\sqrt{49-x^4}}{x} dx &= \int \frac{\sqrt{49-49\sin^2	heta}}{\sqrt{7}\sin	heta} d(\sqrt{7}\sin	heta) \
&= \sqrt{7}\int \frac{\sqrt{\cos^2	heta}}{\sin	heta}d(\sqrt{7}\sin	heta) \
&= \sqrt{7}\int \frac{\cos	heta}{\sin	heta}d(\sqrt{7}\sin	heta) \
&= \sqrt{7}\int \frac{\cos^2	heta+1-1}{\sin	heta}d(\sqrt{7}\sin	heta) \
&= \sqrt{7}\int \frac{\cos^2	heta+1}{\sin	heta}d(\sqrt{7}\sin	heta) - \sqrt{7}\int \frac{1}{\sin	heta}d(\sqrt{7}\sin	heta) \
&= \sqrt{7}\int \frac{\cos^2	heta}{\sin	heta}d(\sqrt{7}\sin	heta) + \sqrt{7}\int \frac{1}{\sin	heta}d(\sqrt{7}\cos	heta) -7\sqrt{7}\int \frac{1}{\sin	heta}d	heta \
&= \sqrt{7}\int \frac{\cos	heta}{\sin	heta}d(\sqrt{7}\cos	heta) + \sqrt{7}\ln|\frac{\sqrt{7}\sin	heta+\sqrt{7}}{\sqrt{7}\cos	heta}| + 7\ln|\sin	heta| + C \
&= \sqrt{7}\ln|\sqrt{49-x^4}+7| + 7\ln|x| + C
\end{align*}
as required. Thus, the answer is (B).

Use a Table of Integrals to Find the Integral ∫49−x4xdx\int \frac { \sqrt { 49 - x ^ { 4 } } } { x } d x∫x49−x4​​dx

Use a Table of Integrals to Find the Integral $\int \frac { \sqrt { 49 - x ^ { 4 } } } { x } d x$