Word Count : 548

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To solve the integral: \[ \int_{0}^{\frac{\pi}{2}} \frac{\cos x}{1 + \cos x + \sin x} \, dx \] we will use a substitution and some algebraic manipulation. --- ### Step 1: Use the substitution \( t = \tan\left(\frac{x}{2}\right) \) The substitution \( t = \tan\left(\frac{x}{2}\right) \) is a standard technique for integrals involving trigonometric functions. Under this substitution: - \( \sin x = \frac{2t}{1 + t^2} \), - \( \cos x = \frac{1 - t^2}{1 + t^2} \), - \( dx = \frac{2}{1 + t^2} \, dt \). When \( x = 0 \), \( t = 0 \), and when \( x = \frac{\pi}{2} \), \( t = 1 ____ _________ ___ _______ _____ ____ _________ _____ ___.
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