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Integral of sinx/(2+sin 2x)............................................................................... please answer

Integral of sinx/(2+sin 2x)............................................................................... please answer

Grade:11

2 Answers

Nishant Vora IIT Patna
askIITians Faculty 2467 Points
4 years ago
Hint : convert this to tan (x/2) form and tjhen solve

jagdish singh singh
173 Points
4 years ago
\hspace{-0.70 cm }$Let $I = \int\frac{\sin x}{2+\sin 2x}dx = \frac{1}{2}\int \frac{(\sin x+\cos x)+(\sin x-\cos x)}{2+\sin 2x}dx$\\\\\\ So $I = \frac{1}{2}\int\frac{\sin x+\cos x}{3-(\sin x-\cos x)^2}+\frac{1}{2}\int\frac{\sin x-\cos x}{1+(\sin x+\cos x)^2}dx$\\\\\\ Now put $(\sin x-\cos x)=t\;,$ Then $(\cos x+\sin x)dx = dt$ in $(1)$\\\\ And Put $(\sin x+\cos x)=u\;,$ Then $(\sin x-\cos x)dx = du$ in $(2)$\\\\
 
\hspace{-0.70 cm }$So $I = \frac{1}{2}\int\frac{1}{(\sqrt{3})^2-t^2}dt-\frac{1}{2}\int\frac{1}{1+u^2}du$\\\\\\ So $I = \frac{1}{4\sqrt{3}}\ln\left|\frac{\sqrt{3}+t}{\sqrt{3}-t}\right|-\frac{1}{2}\tan^{-1}(u)+\mathcal{C}$\\\\\\ So $I = \frac{1}{4\sqrt{3}}\ln\left|\frac{\sqrt{3}+(\sin x-\cos x)}{\sqrt{3}-(\sin x-\cos x)}\right|-\frac{1}{2}\tan^{-1}(\sin x+\cos x)+\mathcal{C}$\\\\\\

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