cos 2r20.(cos x +sin x
The given function is cos ( 2 x ) ( cos x + sin x ) 2 .
The given function can be written as,
cos ( 2 x ) ( cos x + sin x ) 2 = cos ( 2 x ) cos 2 x + sin 2 x + 2 sin x cos x = cos ( 2 x ) 1 + sin 2 x (1)
From (1), we get,
∫ cos ( 2 x ) ( cos x + sin x ) 2 d x = ∫ cos ( 2 x ) 1 + sin ( 2 x ) d x
Put 1 + sin ( 2 x ) = t 2 cos ( 2 x ) d x = d t
Substitute t for 1 + sin ( 2 x ) and d t for 2 cos ( 2 x ) d x in (1),
∫ cos ( 2 x ) ( cos x + sin x ) 2 d x = 1 2 ∫ 1 t d t = 1 2 log | t | + c = 1 2 log | 1 + sin ( 2 x ) | + c = 1 2 log | ( sin ( x ) + cos ( x ) ) 2 | + c = log ( sin x + cos x ) + c
Thus, the integral of the function cos ( 2 x ) ( cos x + sin x ) 2 is log ( sin x + cos x ) + c .
The given function is
The given function can be written as,
From (1), we get,
Put
Substitute
Thus, the integral of the function
