Integrate: $s i n^{2} (2 x + 3)$

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Solution

$\int s i n^{2} (2 x + 3) d x$
Let, 2x+3 = t
$\Rightarrow 2 d x = d t$
$\Rightarrow d x = \frac{d t}{2}$
$∴ \frac{1}{2} \int (s i n^{2} t . d t)$ Now, $s i n^{2} t = \frac{1}{2} (1 - c o s 2 t)$
$= \frac{1}{4} \int (1 - c o s 2 t) d t$
$= \frac{1}{4} [t + \frac{s i n 2 t}{2}] + c$
$= \frac{1}{4} \frac{1}{4} [(2 x + 3) - \frac{s i n (4 x + 6)}{2}] + c$
$= \frac{4 x + 6 - s i n (4 x + 6)}{8} + c$ .

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