1. Polynomial Division: Divide the denominator into the numerator (if needed) to write the integrand as a polynomial plus a proper rational function. 2. Partial Fraction Expansion : Expand the proper rational function using partial fractions. 3. Completing the Square : If any terms involve quadra...
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Step 5: Substitute dx into the IntegralNow substituting dx into the integral: I=∫21+t2⋅1+t24t+4+2t2dt=∫24t+4+2t2dt. Step 6: Factor Out ConstantsFactoring out constants in the denominator gives: I=∫12t2+4t+4dt. Step 7: Complete the SquareCompleting the square for the quadratic...
This implies (see e.g., [30]) that the superlevels of w are balls and, thus, is a ball, completing the proof. Remark 3.7 Theorem 3.6 represents an extension to the double phase case of the so-called Faber-Krahn inequality (cf. [30] for the single phase case). It was firstly...