First slide

Methods of integration

Question

$\int_{}^{} \frac{(2 x^{12} + 5 x^{9}) d x}{{(1 + x^{3} + x^{5})}^{3}} =$

Moderate

A

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

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

$\ln (x^{5} + x^{3} + 1 + \sqrt{(2 x^{12} + 5 x^{9})}) + c$
D

$\ln (x^{5} + x^{3} + 1 + \sqrt{(2 x^{12} - 5 x^{9})}) + c$

Solution

$I = \int_{}^{} \frac{2 x^{12} + 5 x^{9}}{{(1 + x^{3} + x^{5})}^{3}} d x = \int_{}^{} \frac{(5 x^{- 6} + 2 x^{- 3}) d x}{{(1 + x^{- 2} + x^{- 5})}^{3}}$
$Putting 1 + x^{- 2} + x^{- 5} = t, then$
$(- 2 x^{- 3} - 5 x^{- 6}) d x = d t$
$∴ I = - \int_{}^{} \frac{d t}{t^{3}} = \frac{1}{2 t^{2}} = \frac{1}{2 {(1 + x^{- 2} + x^{- 5})}^{2}}$
$= \frac{x^{10}}{2 {(x^{5} + x^{3} + 1)}^{2}} + c$

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