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Question
\[\int\text{ cos x cos 2x cos 3x dx}\]
Sum
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Solution
\[\int\text{ cos x . cos 2x . cos 3x dx}\]
\[ \Rightarrow \frac{1}{2}\int\left[ 2 \cos 2x \cdot \cos x \right] \text{ cos 3x dx}\]
\[ \Rightarrow \frac{1}{2}\int\left[ \text{ cos } \left( 2x + x \right) + \text{ cos } \left( 2x - x \right) \right] \text{ cos 3x dx} ..............\left[ \because 2\text{ cos }A\text{ cos B }= \cos \left( A + B \right) + \text{ cos }\left( A - B \right) \right]\]
\[ \Rightarrow \frac{1}{2}\int\left( \cos3x + \cos x \right) \text{ cos 3x dx }\]
\[ \Rightarrow \frac{1}{2}\int \text{ cos }^2 \text{ 3x dx} + \frac{1}{2}\int\text{ cos } 3x \cdot \text{ cos x dx}\]
\[ \Rightarrow \frac{1}{2}\int\left( \frac{1 + \text{ cos }6x}{2} \right)dx + \frac{1}{4}\int2 \text{ cos 3x} \cdot \text{ cos x dx} ...................\left[ \because \cos 2x = \cos^2 x - 1 \right]\]
\[ \Rightarrow \frac{1}{4}\left[ x + \frac{\sin 6x}{6} \right] + \frac{1}{4}\int\left( \cos 4x + \cos 2x \right)dx\]
\[ \Rightarrow \frac{1}{4}\left[ x + \frac{\sin 6x}{6} \right] + \frac{1}{4}\left[ \frac{\sin 4x}{4} + \frac{\sin 2x}{2} \right] + C\]
\[ \Rightarrow \frac{x}{4} + \frac{\sin 6x}{24} + \frac{\sin 4x}{16} + \frac{\sin 2x}{8} + C\]
\[ \Rightarrow \frac{1}{2}\int\left[ 2 \cos 2x \cdot \cos x \right] \text{ cos 3x dx}\]
\[ \Rightarrow \frac{1}{2}\int\left[ \text{ cos } \left( 2x + x \right) + \text{ cos } \left( 2x - x \right) \right] \text{ cos 3x dx} ..............\left[ \because 2\text{ cos }A\text{ cos B }= \cos \left( A + B \right) + \text{ cos }\left( A - B \right) \right]\]
\[ \Rightarrow \frac{1}{2}\int\left( \cos3x + \cos x \right) \text{ cos 3x dx }\]
\[ \Rightarrow \frac{1}{2}\int \text{ cos }^2 \text{ 3x dx} + \frac{1}{2}\int\text{ cos } 3x \cdot \text{ cos x dx}\]
\[ \Rightarrow \frac{1}{2}\int\left( \frac{1 + \text{ cos }6x}{2} \right)dx + \frac{1}{4}\int2 \text{ cos 3x} \cdot \text{ cos x dx} ...................\left[ \because \cos 2x = \cos^2 x - 1 \right]\]
\[ \Rightarrow \frac{1}{4}\left[ x + \frac{\sin 6x}{6} \right] + \frac{1}{4}\int\left( \cos 4x + \cos 2x \right)dx\]
\[ \Rightarrow \frac{1}{4}\left[ x + \frac{\sin 6x}{6} \right] + \frac{1}{4}\left[ \frac{\sin 4x}{4} + \frac{\sin 2x}{2} \right] + C\]
\[ \Rightarrow \frac{x}{4} + \frac{\sin 6x}{24} + \frac{\sin 4x}{16} + \frac{\sin 2x}{8} + C\]
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