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Q.

$\int \frac{2 x^{12} + 5 x^{9}}{{(x^{4} + x^{3} + 1)}^{3}} d x is equal to$

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a

$\frac{x^{10}}{2 {(1 + x^{3} + x^{5})}^{4}} + c$

b

$\frac{x^{2} + 2 x}{{(x^{5} + x^{3} + 1)}^{4}} + c$

c

$\frac{x^{10}}{2 {(x^{5} + x^{3} + 1)}^{2}} + c$

d

$\frac{2 x^{10}}{2 {(x^{5} + x^{3} + 1)}^{3}} + c$

answer is C.

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Detailed Solution

$\begin{array}{l} I = \int \frac{2 x^{12} + 5 x^{9}}{{(x^{5} + x^{3} + 1)}^{3}} d x \\ = \int \frac{(\frac{2}{x^{3}} + \frac{5}{x^{6}}) d x}{(1 + \frac{1}{x^{2}} + \frac{1}{x^{5}})} \\ = \int \frac{- d t}{t^{3}}, where t = 1 + \frac{1}{x^{2}} + \frac{1}{x^{5}} \\ = \frac{1}{2 t^{2}} + C \\ = \frac{1}{2 {(1 + \frac{1}{x^{2}} + \frac{1}{x^{5}})}^{2}} + c \\ = \frac{x^{10}}{2 {(x^{5} + x^{3} + 1)}^{2}} + c \end{array}$

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