$\int 4 \cos \left(x + \frac{\pi}{6}\right) \cos 2x . \cos\left(\frac{5\pi}{6} + x\right)dx $
- $-\left(x - \frac{\sin4x}{4} - \frac{\sin2x}{2}\right) +c $
- $-\left(x + \frac{\sin4x}{4} - \frac{\sin2x}{2}\right) +c $
- $-\left(x - \frac{\sin4x}{4} + \frac{\sin2x}{2}\right) +c $
- $-\left(x + \frac{\sin4x}{4} + \frac{\sin2x}{2}\right) +c $
The Correct Option is A
Solution and Explanation
$=2 \int\left(\cos (2 x+\pi) \cos \frac{2 \pi}{3}\right) \cos\, 2 x\, d x$
$=2 \int\left(-\cos 2 x-\frac{1}{2}\right) \cos\, 2 x\, d x$
$=\int\left(-2 \cos ^{2} 2 x-\cos 2 x\right) d x$
$=-\int(1+\cos 4 x+\cos 2 x) d x$
$=-x-\frac{\sin 4 x}{x}-\frac{\sin 2 x}{2}+c$
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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C