Word Count : 329

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To evaluate the integral: \[ \int \frac{x^{2} \cot^{-1}(x^{3})}{1 + x^{6}} \, dx \] we will use a substitution method. --- ### Step 1: Substitution Let \( u = x^{3} \). Then, the derivative of \( u \) with respect to \( x \) is: \[ \frac{du}{dx} = 3x^{2} \implies du = 3x^{2} \, dx \implies x^{2} \, dx = \frac{1}{3} \, du. \] --- ### Step 2: Rewrite the Integral Substitute \( u = x^{3} \) and \( x^{2} \, dx = _________ __________ ____ __________ _________ ________ ____ _____.
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