sinx-sin4x-dx-

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 \dots