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Question:

For x² ≠ nπ + 1, n ∈ ℕ (the set of natural numbers), the integral ∫x√(2sin(x²)-sin²(x²))/(2sin(x²)+sin²(x²))dx is equal to (where c is a constant of integration).

loge√√√(1/2sec²(x²))+c

1/2loge√√√(sec²(x²/2))+c

1/2loge√√sec(x²)+c

loge√√√sec(x²/2)+c

Solution:

Put (x²)=t ⇒ 2xdx = dt
∴I = 1/2 ∫√(1-cost)/(1+cost)dt = 1/2 ∫tan(t/2)dt = ln|sec(t/2)| + c
I = ln|sec(x²/2)| + c.