Evaluate the Integral

Problem

(∫_^)(1/(cos(x)−1)*d(x))

Solution

1. Apply a trigonometric identity to rewrite the denominator using the half-angle formula cos(x)=1−2*sin2(x/2)

cos(x)−1=(1−2*sin2(x/2))−1

cos(x)−1=−2*sin2(x/2)

2. Substitute the identity back into the integral.

(∫_^)(1/(−2*sin2(x/2))*d(x))

3. Simplify the integrand using the reciprocal identity 1/sin2(θ)=csc2(θ)

−1/2*(∫_^)(csc2(x/2)*d(x))

4. Integrate using the standard integral (∫_^)(csc2(u)*d(u))=−cot(u)+C Here, u=x/2 and d(u)=1/2*d(x) so d(x)=2*d(u)

−1/2*(2*(−cot(x/2)))+C

5. Simplify the final expression by canceling the constants.

cot(x/2)+C

Final Answer

(∫_^)(1/(cos(x)−1)*d(x))=cot(x/2)+C

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