Evaluate the Integral

Problem

(∫_^)(1/(1+cos(x))*d(x))

Solution

1. Apply a trigonometric identity to rewrite the denominator using the half-angle formula cos(x)=2*cos2(x/2)−1

2. Simplify the expression by substituting the identity into the integral.

(∫_^)(1/(1+(2*cos2(x/2)−1))*d(x))

3. Cancel the constants in the denominator.

(∫_^)(1/(2*cos2(x/2))*d(x))

4. Rewrite the integrand using the reciprocal identity 1/cos2(θ)=sec2(θ)

1/2*(∫_^)(sec2(x/2)*d(x))

5. Integrate using the standard integral (∫_^)(sec2(u)*d(u))=tan(u)+C applying the chain rule for the linear argument x/2

1/2⋅tan(x/2)/1/2+C

6. Simplify the coefficients to reach the final form.

tan(x/2)+C

Final Answer

(∫_^)(1/(1+cos(x))*d(x))=tan(x/2)+C

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