Prove that 
sin m x cos n x dx = 
+ 
sin m+2 x cos n x dx
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol. Here, the smaller of the indices of sin x and cos x in the two integrals to be connected are m and n respectively. So, let
P = sin m+1 cos n+1 x
= 
= (m + 1) sin m x cos n+2 x – (n + 1) sin m+2 x cos n x
⇒ 
= (m + 1) sin m x cos n x. cos 2 x – (n + 1) sin m+2 x cos n x
⇒ 
= (m + 1) sin m x cos n x (1 –sin 2 x) – (n + 1) sin m+2 x cos n x
⇒ 
= (m + 1) sin m x cos n x – (m +1 ) sin m+2
x cos n x –(n + 1) sin m+2 x cos n x
⇒ 
= (m + 1) sin m x cos n x – (m + n + 2) sin m+2 x cos n x
Integrating both sides with respect to x, we get
P = (m +1) 
sin m x cos n x dx – ( m + n + 2) 
sin m+2 x cos n x dx
∴ (m + 1) 
sin m x cos n x dx = P + (m + n + 2) 
sin m+2 x cos n x dx
⇒ 
sin m x cos n x dx = 
+ 
sin m+2 x cos n x dx
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