integrate (sin^2 x+1)dx the limit is from -pi/2 to pi/2
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sin(x)^2 + 1 =>
1 - cos(x)^2 + 1 =>
2 - cos(x)^2
(2 - cos(x)^2) * dx
x = t/2
dx = dt / 2
Now, cos(t/2) = sqrt((1 + cos(t)) / 2)
(2 - (1/2) * (1 + cos(t))) * (1/2) * dt =>
(1/2) * (1/2) * (4 - 1 - cos(t)) * dt =>
(1/4) * (3 - cos(t)) * dt
Integrate:
(1/4) * (3t - sin(t)) + C =>
(1/4) * (3 * (2x) - sin(2x)) + C =>
(1/4) * (6x - 2sin(x)cos(x)) + C =>
(1/2) * (3x - sin(x)cos(x)) + C
From -pi/2 to pi/2
(1/2) * (3 * (pi/2) - sin(pi/2)cos(pi/2)) - (1/2) * (3 * (-pi/2) - sin(-pi/2) * cos(-pi/2)) =>
(1/2) * (3pi/2 - 1 * 0)) - (1/2) * (-3pi/2 - (-1) * 0)) =>
(1/2) * (3pi/2) + (1/2) * (3pi/2) =>
3pi/2
1 - cos(x)^2 + 1 =>
2 - cos(x)^2
(2 - cos(x)^2) * dx
x = t/2
dx = dt / 2
Now, cos(t/2) = sqrt((1 + cos(t)) / 2)
(2 - (1/2) * (1 + cos(t))) * (1/2) * dt =>
(1/2) * (1/2) * (4 - 1 - cos(t)) * dt =>
(1/4) * (3 - cos(t)) * dt
Integrate:
(1/4) * (3t - sin(t)) + C =>
(1/4) * (3 * (2x) - sin(2x)) + C =>
(1/4) * (6x - 2sin(x)cos(x)) + C =>
(1/2) * (3x - sin(x)cos(x)) + C
From -pi/2 to pi/2
(1/2) * (3 * (pi/2) - sin(pi/2)cos(pi/2)) - (1/2) * (3 * (-pi/2) - sin(-pi/2) * cos(-pi/2)) =>
(1/2) * (3pi/2 - 1 * 0)) - (1/2) * (-3pi/2 - (-1) * 0)) =>
(1/2) * (3pi/2) + (1/2) * (3pi/2) =>
3pi/2
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sin^2 x + 1 = 3/2 - (1/2)cos(2x) , then it's easy to integrate
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sin^2 x + 1 = 3/2 - (1/2)cos(2x)