Word Count : 725

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To evaluate the integral: \[ \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{1+\tan x} \, dx \] we can use a substitution to simplify the integral. Here's a step-by-step solution: --- ### Step 1: Use the substitution \( u = \tan x \) Let \( u = \tan x \). Then: \[ du = \sec^2 x \, dx \] and \[ dx = \frac{du}{\sec^2 x}. \] From the identity \( \sec^2 x = 1 + \tan^2 x \), we have: \[ dx = \frac{du}{1 + u^2}. \] The integral becomes: \[ \int \frac{1}{1 + u} \cdot \frac{du}{1 + u^2}. \] --- ### Step 2: Partial fraction decomposition We decompose the integrand: \[ \frac{1}{(1 + u)(1 + u^2)} = \frac{A}{1 + u} + \frac{Bu + C}{1 + u^2}. \] Multiply through by \( (1 + u)(1 + u^2) \): \[ 1 = A(1 + u^2) + (Bu + C)(1 + u). \] ___ __________ _________ _____ _______ __________.
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