First slide

Methods of integration

Question

$\int \frac{3 x^{2} + 2 x}{x^{6} + 2 x^{5} + x^{4} + 2 x^{3} + 2 x^{2} + 5} d x =$

Moderate

A

$\frac{1}{4} \tan^{- 1} [\frac{x^{3} + x^{2} + 1}{2}] + c$
B

$\frac{1}{2} \tan^{- 1} [\frac{x^{3} + x^{2} + 1}{2}] + c$
C

$\sin^{- 1} [\frac{x^{3} + x^{2} + 1}{2}] + c$
D

$\frac{1}{2} \tan^{- 1} [\frac{x^{3} + x^{2}}{2}] + c$

Solution

$\int \frac{3 x^{2} + 2 x}{{(x^{3} + x^{2} + 1)}^{2} + 2^{2}} d x$
Let $x^{3} + x^{2} + 1 = t$

$\int \frac{d t}{t^{2} + 2^{2}} = \frac{1}{2} \tan^{- 1} (\frac{t}{2}) + c$

$= \frac{1}{2} \tan^{- 1} [\frac{x^{3} + x^{2} + 1}{2}] + c$

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