Question

Let $I_1=\underset{-2}{\overset{2}{\int }}\frac{x^6+3x^5+7x^4}{x^4+2}dx$ and $I_2=\underset{-3}{\overset{1}{\int }}\frac{2(x+1{)}^2+11(x+1)+14}{(x+1{)}^4+2}dx$,
then the value of $I_1+I_2$ is

• A. $8$
• B. $\frac{200}{3}$
• C. $\frac{100}{3}$
• D. $0$

Answer 1

The correct option is C $\frac{100}{3}$

In $I_2$, put $x+1=t,dx=dt$, then
$I_2=\underset{-2}{\overset{2}{\int }}\frac{2t^2+11t+14}{t^4+2}dt=\underset{-2}{\overset{2}{\int }}\frac{2x^2+11x+14}{x^4+2}dx$
$\therefore I_1+I_2=\underset{-2}{\overset{2}{\int }}\frac{x^6+3x^5+7x^4+2x^2+11x+14}{x^4+2}dx$
$=\underset{-2}{\overset{2}{\int }}\frac{(x^2+3x+7)(x^4+2)+5x}{x^4+2}dx$
$=\underset{-2}{\overset{2}{\int }}(x^2+3x+7)dx+5\underset{-2}{\overset{2}{\int }}\frac{x}{x^4+2}dx$
$=2\underset{0}{\overset{2}{\int }}(x^2+7)dx+0+0=2{[\frac{x^3}{3}+7x]}_0^2=\frac{100}{3}$
$\because \underset{-a}{\overset{a}{\int }}f\left(x\right)dx=\{\begin{array}{cc}0,& \text{if}f\left(x\right)\text{is odd}\\ 2\underset{0}{\overset{a}{\int }}f\left(x\right)dx,& \text{if}f\left(x\right)\text{is even}\end{array}$