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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS3 Marks
Question
Find the integrals of the functions in Exercises:
$\sin3\text{x}\cos4\text{x}$
✓

Answer

It is know that, $\sin\text{A}\cos\text{B}=\frac{1}{2}\big\{\sin(\text{A+B})+\sin(\text{A}-\text{B}) \big\} $
$\therefore\int\sin3\text{x}\cos4\text{x dx}=\frac{1}{2}\int\big\{\sin(3\text{x}+4\text{x})+\sin(\text{3x}-\text{4x}) \big\}\text{ dx} $
$=\frac{1}{2}\int\big\{\sin\text{7x}+\sin(-\text{x}) \big\}\text{ dx} $
$=\frac{1}{2}\int\big\{\sin\text{7x}-\sin\text{x} \big\}\text{ dx} $
$=\frac{1}{2}\int\sin\text{7x dx}-\frac{1}{2}\int\sin\text{x} \text{ dx} $
$=\frac{1}{2}\bigg(\frac{-\cos7\text{x}}{7}\bigg)-\frac{1}{2}(-\cos\text{x}) +\text{ C} $
$=\frac{-\cos7\text{x}}{14}+\frac{\cos\text{x}}{2}+\text{C} $

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