Question

$\displaystyle \int \frac{dx}{1+\cos x}$ is equal to

• A. $\dfrac{1}{2}\tan\dfrac{x}{2}+C$
• B. $-\dfrac{1}{2}\cot\dfrac{x}{2}+C$
• C. $\cot\dfrac{x}{2}+C$
• D. $\tan\dfrac{x}{2}+C$

Hint:

Whenever integrals contain \(1+\cos x\) or \(1-\cos x\), convert them using half-angle identities such as \(1+\cos x=2\cos^2(x/2)\). This simplifies the integral immediately.

Answer 1

The Correct Option is D

To solve the integral \(\int \frac{dx}{1+\cos x}\), we will make use of the trigonometric identity and substitution method. 

1. Trigonometric Identity:
2. Rewrite the Integral:
   • \(\displaystyle \int \frac{dx}{1 + \cos x} = \int \frac{dx}{1 + (1 - 2\sin^2\left(\frac{x}{2}\right))}\)
   • = \(\int \frac{dx}{2 - 2\sin^2\left(\frac{x}{2}\right)}\)
   • = \(\int \frac{dx}{2(1 - \sin^2\left(\frac{x}{2}\right))}\)
   • = \(\int \frac{dx}{2\cos^2\left(\frac{x}{2}\right)} = \frac{1}{2} \int \sec^2\left(\frac{x}{2}\right) dx\)
3. Substitution:
   • The integral becomes: \(\frac{1}{2} \int \sec^2(t) \cdot 2 \, dt = \int \sec^2(t) \, dt\)
4. Integration:
   • \(\int \sec^2(t) \, dt = \tan(t) + C\)
   • Since \(t = \frac{x}{2}\), we substitute back to get \(\tan\left(\frac{x}{2}\right) + C\)

Therefore, the solution to the integral \(\int \frac{dx}{1+\cos x}\) is \(\tan\left(\frac{x}{2}\right) + C\).