Calculate Integrals: x= 2 cos â - cos 2 â -1, y= 2 sin â - sin 2 â?
I need the procedure.
The answer is: 6ð
Thank you!
To calculate the integrals of x = 2cos(α) - cos^2(α) - 1 and y = 2sin(α) - sin^2(α), we can use the trigonometric identities and integral properties.
1. Start by rewriting the expressions using trigonometric identities:
x = 2cos(α) - (1/2)(1 + cos(2α)) - 1
y = 2sin(α) - (1/2)(1 - cos(2α))
2. Simplify the expressions further:
x = 2cos(α) - (1/2) - (1/2)cos(2α) - 1
y = 2sin(α) - (1/2) + (1/2)cos(2α)
3. Separate the terms with respect to α:
x = - (1/2) - 1 + 2cos(α) - (1/2)cos(2α)
y = - (1/2) + 2sin(α) + (1/2)cos(2α)
4. Now, we can integrate each term separately:
∫x dα = ∫(-1/2) dα + ∫(-1) dα + ∫2cos(α) dα - ∫(1/2)cos(2α) dα
∫y dα = ∫(-1/2) dα + ∫2sin(α) dα + ∫(1/2)cos(2α) dα
5. Integrate each term using the appropriate rules:
∫(-1/2) dα = (-1/2)α + C
∫(-1) dα = -α + C
∫2cos(α) dα = 2sin(α) + C
∫(1/2)cos(2α) dα = (1/4)sin(2α) + C
∫(-1/2) dα = (-1/2)α + C
∫2sin(α) dα = -2cos(α) + C
∫(1/2)cos(2α) dα = (1/4)sin(2α) + C
6. Substitute the limits of integration if provided and evaluate the integrals:
∫x dα = (-1/2)α - α + 2sin(α) + (1/4)sin(2α) + C
∫y dα = (-1/2)α - 2cos(α) + (1/4)sin(2α) + C
7. The answer states that x = 6π, which means the integral of x with respect to α from some lower limit a to some upper limit b equals 6π (C is the constant of integration). However, without provided limits, we cannot determine the exact value of C or evaluate the integrals fully.
In conclusion, the procedure to calculate the integrals of x and y is explained above, but without specific limits of integration, we cannot calculate the exact values of ∫x dα and ∫y dα.