Class 12 Maths · Integrals · Long Answer #164

Question

Evaluate $\int \frac{1 - \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right) \cos{\left(x \right)}}\,dx$. Work on each connected open interval where the real integrand is defined.

Accepted Answer

Final Answer: $\log{\left(\left|{\tan{\left(x \right)} + \sec{\left(x \right)}}\right| \right)} - 2 \tan{\left(\frac{x}{2} \right)}$ + C Step 1: Decompose (1−cos x)/[cos x(1+cos x)]=sec x−2/(1+cos x). Step 2: Use 1+cos x=2cos²(x/2), so the second term is −sec²(x/2). Step 3: Integrate sec x to ln|sec x+tan x| and −sec²(x/2) to −2tan(x/2). Step 4: Therefore the required integral is $\log{\left(\left|{\tan{\left(x \right)} + \sec{\left(x \right)}}\right| \right)} - 2 \tan{\left(\frac{x}{2} \right)}$ + C.

Class 12 Maths · Integrals · Long Answer #164

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