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int(dx)/(2+cos x)

55) ∫dx2+cos⁡x \int \frac{d x}{2+\cos x}

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Full solution

Q. 55) ∫dx2+cos⁡x \int \frac{d x}{2+\cos x}
  1. Trig Identity Rewrite: Rewrite the integral using a trigonometric identity for easier integration.\newlineWe use the identity: cos⁡(x)=1−2sin⁡2(x2) \cos(x) = 1 - 2\sin^2(\frac{x}{2}) .\newlineSo, 2+cos⁡(x)=3−2sin⁡2(x2) 2 + \cos(x) = 3 - 2\sin^2(\frac{x}{2}) .\newlineRewrite the integral: ∫dx3−2sin⁡2(x2) \int \frac{dx}{3 - 2\sin^2(\frac{x}{2})} .
  2. Substitution with u: Substitute u=sin⁡(x2) u = \sin(\frac{x}{2}) , then du=12cos⁡(x2)dx du = \frac{1}{2}\cos(\frac{x}{2})dx .\newlineSolving for dx dx , we get dx=2ducos⁡(x2) dx = 2\frac{du}{\cos(\frac{x}{2})} .

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