First slide

Methods of integration

Question

$\int (x^{6} + x^{4} + x^{2}) \sqrt{2 x^{4} + 3 x^{2} + 6} d x$

Moderate

A

$\frac{1}{18} {(2 x^{6} + 3 x^{4} + 6 x^{2})}^{3 / 2} + C$
B

$\frac{1}{12} {(2 x^{6} + 3 x^{4} + 6 x^{2})}^{3 / 2} + C$
C

$\frac{1}{18} {(2 x^{6} + 3 x^{4} + 6 x^{2})}^{2 / 3} + C$
D

$\frac{1}{12} {(2 x^{6} + 3 x^{4} + 6 x^{2})}^{2 / 3} + C$

Solution

$\begin{array}{l} I = \int (x^{6} + x^{4} + x^{2}) \sqrt{2 x^{4} + 3 x^{2} + 6} \\ = \int (x^{5} + x^{3} + x) \sqrt{2 x^{6} + 3 x^{4} + 6 x^{2}} dx \\ Let 2 x^{6} + 3 x^{4} + 6 x^{2} = t^{2} \\ ∴ 12 (x^{5} + x^{3} + x) d x = 2 t dt \\ ∴ I = \frac{1}{12} \int 2 t^{2} d t = \frac{1}{18} {(2 x^{6} + 3 x^{4} + 6 x^{2})}^{3 / 2} + c \end{array}$

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