∫((sinx)/(sin4x))dx


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Question Number 49187 by Rahul kharade last updated on 04/Dec/18

$\int \frac{s i n x}{s i n 4 x} d x$
	Answered by MJS last updated on 04/Dec/18

	$\int \frac{\sin x}{\sin 4 x} d x =$ $[t = \tan \frac{x}{2} \to d x = \frac{2 d t}{t^{2} + 1}]$ $= - \frac{1}{2} \int \frac{{(t^{2} + 1)}^{2}}{(t - 1) (t + 1) (t - 1 - \sqrt{2}) (t - 1 + \sqrt{2}) (t + 1 - \sqrt{2}) (t + 1 + \sqrt{2})} d t =$ $= - \frac{1}{2} \int (- \frac{1}{2 (t - 1)} + \frac{1}{2 (t + 1)} + \frac{\sqrt{2}}{4 (t - 1 - \sqrt{2})} - \frac{\sqrt{2}}{4 (t - 1 + \sqrt{2})} + \frac{\sqrt{2}}{4 (t + 1 - \sqrt{2})} - \frac{\sqrt{2}}{4 (t + 1 + \sqrt{2})}) =$ $[now solve with formula \int \frac{d t}{t + a} = \ln ∣ t + a ∣]$ $= \frac{\sqrt{2}}{8} \ln ∣ \frac{t^{2} + 2 \sqrt{2} t + 1}{t^{2} - 2 \sqrt{2} t + 1} ∣ + \frac{1}{4} \ln ∣ \frac{t - 1}{t + 1} ∣ =$ $= \frac{\sqrt{2}}{8} \ln ∣ \frac{1 + \sqrt{2} \sin x}{1 - \sqrt{2} \sin x} ∣ + \frac{1}{4} \ln ∣ \frac{1}{\tan (\frac{x}{2} + \frac{π}{4})} ∣ + C$
	Answered by tanmay.chaudhury50@gmail.com last updated on 04/Dec/18

	$\int \frac{s i n x}{2 s i n 2 x c o s 2 x} d x$ $\int \frac{s i n x}{2 (2 s i n x c o s x) (c o s 2 x)} d x$ $\int \frac{d x}{4 c o s x c o s 2 x} d x$ $\int \frac{c o s x d x}{4 (1 - {s i n}^{2} x) (1 - 2 {s i n}^{2} x)}$ $\int \frac{d t}{4 (1 - t^{2}) (1 - 2 t^{2})} t = s i n x$ $\frac{1}{4} \int \frac{(2 - 2 t^{2}) - (1 - 2 t^{2})}{(1 - t^{2}) (1 - 2 t^{2})} d t$ $\frac{1}{4} \int \frac{2 d t}{1 - 2 t^{2}} - \frac{1}{4} \int \frac{d t}{1 - t^{2}}$ $\frac{1}{2} \int \frac{d t}{2 (\frac{1}{2} - t^{2})} - \frac{1}{4} \int \frac{d t}{1 - t^{2}}$ $\frac{1}{4} \int \frac{d t}{{(\frac{1}{\sqrt{2}})}^{2} - t^{2}} - \frac{1}{4} \int \frac{d t}{1 - t^{2}}$ $n o w u s e f o r m u l a \int \frac{d x}{a^{2} - x^{2}} = \frac{1}{2 a} l n (\frac{a + x}{a - x}) + c$ $s o \frac{1}{4} \times \frac{1}{2 \times \frac{1}{\sqrt{2}}} l n (\frac{\frac{1}{\sqrt{2}} + t}{\frac{1}{\sqrt{2}} - t}) + \frac{1}{4} \times \frac{1}{2} l n (\frac{1 + t}{1 - t}) + c$ $\frac{1}{4 \sqrt{2}} l n (\frac{1 + \sqrt{2} s i n x}{1 - \sqrt{2} s i n x}) + \frac{1}{8} l n (\frac{1 + s i n x}{1 - s i n x}) + c$ $p l s c h e c k . . .$
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