Find the integrals of the functions in Exercises: 4x 8x - Vidyadip

Question

Find the integrals of the functions in Exercises:
$\sin4\text{x}\sin8\text{x}$

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Answer

It is known that,

$\sin\text{A}\sin\text{B}=\frac{1}{2}\cos(\text{A}-\text{B})-\cos(\text{A}+\text{B})$

$=\int\bigg\{\frac{1}{2}\cos(4\text{x}-8\text{x})-\cos(4\text{x}+8\text{x})\bigg\}\text{ dx}$

$=\frac{1}{2}\int(\cos(-4\text{x})-\cos12\text{x})\text{dx}$

$=\frac{1}{2}\int\big\{(\cos4\text{x}-\cos12\text{x}\big\}\text{dx}$

$=\frac{1}{2}\bigg[\frac{\sin4\text{x}}{4}-\frac{\sin12\text{x}}{12}\bigg]+\text{C}$

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