To solve the integral ∫(sin(x)/4 + cos(x)/4)^2 dx, we can expand the square and simplify the expression as follows:
(sin(x)/4 + cos(x)/4)^2= (sin^2(x)/16 + sin(x)cos(x)/8 + cos^2(x)/16)= (1/16)sin^2(x) + (1/8)sin(x)cos(x) + (1/16)cos^2(x)= (1/16)(sin^2(x) + 2sin(x)cos(x) + cos^2(x))= (1/16)(sin(x) + cos(x))^2
Therefore, the integral simplifies to:
∫(sin(x)/4 + cos(x)/4)^2 dx= ∫(1/16)(sin(x) + cos(x))^2 dx= (1/16)∫(sin(x) + cos(x))^2 dx
We can further simplify this by expanding the square and using trigonometric identities.
(sin(x) + cos(x))^2 = sin^2(x) + 2sin(x)cos(x) + cos^2(x) = 1 + sin(2x)
Therefore, the integral becomes:
(1/16)∫(sin(x) + cos(x))^2 dx= (1/16)∫(1 + sin(2x)) dx= (1/16)x + (1/32)cos(2x) + C
where C is the constant of integration.
So, ∫(sin(x)/4 + cos(x)/4)^2 dx = (1/16)x + (1/32)cos(2x) + C + D
where C and D are constants of integration.
To solve the integral ∫(sin(x)/4 + cos(x)/4)^2 dx, we can expand the square and simplify the expression as follows:
(sin(x)/4 + cos(x)/4)^2
= (sin^2(x)/16 + sin(x)cos(x)/8 + cos^2(x)/16)
= (1/16)sin^2(x) + (1/8)sin(x)cos(x) + (1/16)cos^2(x)
= (1/16)(sin^2(x) + 2sin(x)cos(x) + cos^2(x))
= (1/16)(sin(x) + cos(x))^2
Therefore, the integral simplifies to:
∫(sin(x)/4 + cos(x)/4)^2 dx
= ∫(1/16)(sin(x) + cos(x))^2 dx
= (1/16)∫(sin(x) + cos(x))^2 dx
We can further simplify this by expanding the square and using trigonometric identities.
(sin(x) + cos(x))^2 = sin^2(x) + 2sin(x)cos(x) + cos^2(x) = 1 + sin(2x)
Therefore, the integral becomes:
(1/16)∫(sin(x) + cos(x))^2 dx
= (1/16)∫(1 + sin(2x)) dx
= (1/16)x + (1/32)cos(2x) + C
where C is the constant of integration.
So, ∫(sin(x)/4 + cos(x)/4)^2 dx = (1/16)x + (1/32)cos(2x) + C + D
where C and D are constants of integration.