Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int _ { 3 } ^ { 7 } \frac { 4 x ^ { 3 } - 4 x } { x ^ { 4 } - 16 } d x$$
A) -3.4
B) 3.4
C) 3.8
D) 6.81

Question

Quizplus · Accepted Answer

First, factor the denominator: $x^4-16=(x^2-4)(x^2+4)=(x-2)(x+2)(x^2+4)$.
Then, write the integrand as a sum of partial fractions: 
$$\frac{4x^3-4x}{x^4-16}=\frac{A}{x-2}+\frac{B}{x+2}+\frac{Cx+D}{x^2+4}$$
Multiplying both sides by the denominator, we get:
$$4x^3-4x=A(x+2)(x^2+4)+B(x-2)(x^2+4)+(Cx+D)(x^2-4)$$
Setting $x=2$, we get $-32=24A\Rightarrow A=-\frac{4}{3}$. 
Setting $x=-2$, we get $32=-24B\Rightarrow B=-\frac{4}{3}$.
Expanding the right side and equating coefficients of the like terms, we get:
\begin{align*}
4x^3&=(A+B)x^3+Cx^3\
-4x&=(2A-2B+D)x\
0&=(2A+2B) x+4D
\end{align*}
From the first equation, we get $C=8$. From the second equation, we get $D=2A-2B$. Substituting into the third equation and using the values of $A$ and $B$, we get $0=0$, meaning that the third equation holds for any values of $A$ and $B$. Therefore, we can choose any convenient values for $A$ and $B$ to calculate $D$. For example, setting $A=1$ and $B=0$, we get $D=2A-2B=2$. Thus, we have:
$$\frac{4x^3-4x}{x^4-16}=\frac{-\frac{4}{3}}{x-2}+\frac{-\frac{4}{3}}{x+2}+\frac{8x+2}{x^2+4}$$
Integrating each term separately, we get:
\begin{align*}
\int_3^7\frac{-\frac{4}{3}}{x-2}dx&=-\frac{4}{3}(\ln|{x-2}|)\bigg|_3^7=\frac{4}{3}\ln\left(\frac{7-2}{3-2}ight)=\frac{4}{3}\ln\frac{5}{3}\
\int_3^7\frac{-\frac{4}{3}}{x+2}dx&=-\frac{4}{3}(\ln|{x+2}|)\bigg|_3^7=-\frac{4}{3}\ln\left(\frac{7+2}{3+2}ight)=\frac{4}{3}\ln\frac{5}{9}\
\int_3^7\frac{8x+2}{x^2+4}dx&=4(\ln|{x^2+4}|)\bigg|_3^7=4\ln\left(\frac{7^2+4}{3^2+4}ight)=4\ln\frac{41}{13}
\end{align*}
Adding the three integrals, we get:
$$\int _ { 3 } ^ { 7 } \frac { 4 x ^ { 3 } - 4 x } { x ^ { 4 } - 16 } d x=\frac{4}{3}\ln\frac{5}{3}+\frac{4}{3}\ln\frac{5}{9}+4\ln\frac{41}{13}=4\ln\frac{41}{27}\approx 3.4$$
Therefore, the answer is $\boxed{	extbf{(B)}}$.

Express the Integrand as a Sum of Partial Fractions and Evaluate