integral of SQRT (2 + 2sinT) dT
show your work please
show your work please
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∫√(2 + 2*sin(T)) dT
√2 * ∫√(1 + sin(T)) dT
Recall that sin(2T) = 2*sin(T)*cos(T). Then sin(T) = 2*sin(T/2)*cos(T/2)
√2 * ∫√(1 + 2*sin(T/2)*cos(T/2)) dT
From knowing that sin²(T/2) + cos²(T/2) = 1, the integral becomes:
√2 * ∫√(sin²(T/2) + 2*sin(T/2)cos(T/2) + cos²(T/2)) dT
√2 * ∫√[sin(T/2) + cos(T/2)]² dt
√2 * ∫(sin(T/2) + cos(T/2)) dt
√2 * [-2*cos(T/2) + 2*sin(T/2)] + C
Answer: 2√2*(sin(T/2) - cos(T/2)) + C
√2 * ∫√(1 + sin(T)) dT
Recall that sin(2T) = 2*sin(T)*cos(T). Then sin(T) = 2*sin(T/2)*cos(T/2)
√2 * ∫√(1 + 2*sin(T/2)*cos(T/2)) dT
From knowing that sin²(T/2) + cos²(T/2) = 1, the integral becomes:
√2 * ∫√(sin²(T/2) + 2*sin(T/2)cos(T/2) + cos²(T/2)) dT
√2 * ∫√[sin(T/2) + cos(T/2)]² dt
√2 * ∫(sin(T/2) + cos(T/2)) dt
√2 * [-2*cos(T/2) + 2*sin(T/2)] + C
Answer: 2√2*(sin(T/2) - cos(T/2)) + C