Evaluate the following: ( 5x+ 4x) 1-2 3x dx

Question

Evaluate the following:
$\int\frac{(\cos5\text{x}+\cos4\text{x})}{1-2\cos3\text{x}}\text{dx}$

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Answer

Let $\text{I}=\int\frac{(\cos5\text{x}+\cos4\text{x})}{1-2\cos3\text{x}}\text{dx}$
$=\int\frac{2\cos\frac{9\text{x}}{2}\cdot\cos\frac{\text{x}}{2}}{1-2\Big(2\cos^2\frac{3\text{x}}{2}-1\Big)}\text{dx}$
$=-\int\frac{2\cos\frac{9\text{x}}{2}\cdot\cos\frac{\text{x}}{2}}{4\cos^2\frac{3\text{x}}{2}-3}\text{dx}$
$=-\int\frac{2\cos\frac{9\text{x}}{2}\cdot\cos\frac{\text{x}}{2}\cdot\cos\frac{3\text{x}}{2}}{4\cos^3\frac{3\text{x}}{2}-3\cos\frac{3\text{x}}{2}}\text{dx}$ $\big[\because\cos3\text{x}=4\cos^3\text{x}-3\cos\text{x}\big]$
$=-\int\frac{2\cos\frac{9\text{x}}{2}\cdot\cos\frac{\text{x}}{2}\cdot\cos\frac{3\text{x}}{2}}{\cos3\cdot\frac{3\text{x}}{2}}\text{dx}$
$=-\int2\cos\frac{3\text{x}}{2}\cdot\cos\frac{\text{x}}{2}\text{dx}$
$=\int\bigg\{\cos\Big(\frac{3\text{x}}{2}+\frac{\text{x}}{2}\Big)+\cos\Big(\frac{3\text{x}}{2}-\frac{\text{x}}{2}\Big)\bigg\}\text{dx}$
$=-\int(\cos2\text{x}+\cos\text{x})\text{dx}=-\frac{1}{2}\sin2\text{x}-\sin\text{x}+\text{C}$

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