Question

The value of$\int \frac{x^5+x^4+4x^3+4x^2+4x+4}{(x^2+2{)}^3}dx$ is equal to

• A. $\frac{4x-3}{(x^2+1{)}^2}+\frac{3}{8}\frac{x}{x^2+2}+\frac{1}{\sqrt{2}}{\tan }^{-1}\frac{x}{\sqrt{2}+c}+c$
• B. $\frac{1}{12}\frac{2x-3}{(x^2+1{)}^2}+\frac{3}{16}\frac{x}{x^2+2}+\frac{3}{16\sqrt{2}}{\tan }^{-1}\frac{x}{\sqrt{2}}$
• C. $\frac{2x-3}{(x^2+1{)}^2}+\frac{3}{8}\frac{x}{x^2+2}+\frac{1}{2\sqrt{2}}{\tan }^{-1}\frac{x}{\sqrt{2}}+c$
• D. $Noneofthese$

Answer 1

The correct option is D $Noneofthese$

$\int \frac{x^5+x^4+4x^3+4x^2+4x+4}{(x^2+2{)}^3}dx$
$=\int \frac{(x^5+2x^3)+(x^4-2x^2)+(2x^3+4x)+(2x^2+4)}{(x^2+2{)}^3}dx$
$=\int \frac{x^3}{(x^2+2{)}^2}dx+\int \frac{x^2}{(x^2+2{)}^2}dx+2\int \frac{x}{(x^2+2{)}^2}dx+2\int \frac{1}{(x^2+2{)}^2}dx$
$=\int \frac{x^3+x^2+2x+2}{(x^2+2{)}^2}dx=\int \frac{x}{(x^2+2)}dx+\int \frac{1}{(x^2+2)}dx$
$=\int \frac{x}{(x^2+2)}dx+\int \frac{1}{x^2+2}dx$
$=\frac{1}{2}\int \frac{d{x}^2}{x^2+2}+\int \frac{1}{x^2+2}dx$
$=\frac{1}{2}log\mid x^2+2\mid +\frac{1}{\sqrt{2}}ta{n}^{-1}\left(\frac{x}{\sqrt{2}}\right)+c.$