∫ (b,0) a²sin²(πx/b) / 2
where b = upper limit
where 0 = lower limit
answer is a²b/4
where b = upper limit
where 0 = lower limit
answer is a²b/4
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The a^2/2 is just a constant and can be pulled out of the integral
a²/2 ∫ (b,0) sin²(πx/b)
Use the half angle formula
a²/2 ∫ (b,0) [1-Cos(2πx/b)]/2
Pull out the extra 1/2 u picked up when u used the half angle formula
a²/4 ∫ (b,0) [1-Cos(2πx/b)]
Now the integral is easy
a²/4 [(b - b/2π Sin(2π)) - (0-b/2π Sin(0))] = a²b/4
a²/2 ∫ (b,0) sin²(πx/b)
Use the half angle formula
a²/2 ∫ (b,0) [1-Cos(2πx/b)]/2
Pull out the extra 1/2 u picked up when u used the half angle formula
a²/4 ∫ (b,0) [1-Cos(2πx/b)]
Now the integral is easy
a²/4 [(b - b/2π Sin(2π)) - (0-b/2π Sin(0))] = a²b/4