integral from 0 to pi of 9+6cos((pi/25)y)+cos^2((pi/25)y)
To evaluate the integral from 0 to π of 9 + 6cos((π/25)y) + cos^2((π/25)y), we can follow these steps:
Step 1: Simplify the expression
The given integral can be written as:
∫(0 to π) (9 + 6cos((π/25)y) + cos^2((π/25)y)) dy
Step 2: Expand cos^2((π/25)y)
Using the identity cos^2θ = (1 + cos(2θ))/2, we can expand the squared cosine term:
∫(0 to π) (9 + 6cos((π/25)y) + (1 + cos((π/25)y))/2) dy
Step 3: Distribute the 1/2 coefficient
Now, distribute the 1/2 coefficient to the expanded term:
∫(0 to π) (9 + 6cos((π/25)y) + 1/2 + cos((π/25)y)/2) dy
Step 4: Rearrange the terms
Rearrange the terms to group the constants and cosine terms:
∫(0 to π) (9 + 1/2 + 6cos((π/25)y) + cos((π/25)y)/2) dy
Step 5: Integrate each term separately
Integrate each term separately. Since the integral of a constant is simply the constant times the interval, and the integral of cos(ax) is sin(ax)/a, we can evaluate the integral:
= [9y + (1/2)y + 6(25/π)sin((π/25)y) + (2/π)sin((π/25)y)] evaluated from 0 to π
Step 6: Evaluate the integral over the given limits
Now, substitute π into the equation and evaluate each term:
= [9π + (1/2)π + 6(25/π)sin((π/25)π) + (2/π)sin((π/25)π)] - [0 + 0 + 6(25/π)sin(0) + (2/π)sin(0)]
Step 7: Simplify the trigonometric terms
By using the trigonometric identities sin(π) = 0 and sin(0) = 0, we can simplify the equation further:
= [9π + (1/2)π + 0 + 0] - [0 + 0 + 0 + 0]
Step 8: Final result
The final result of the integral from 0 to π of 9 + 6cos((π/25)y) + cos^2((π/25)y) is:
= 9π + (1/2)π
= (18π + π)/2
= (19π)/2
Therefore, the value of the given integral is (19π)/2.