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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

\int_{0}^{\infty} \frac{6}{e^{3x} + 6e^{2x} + 11e^{x} + 6} dx

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Explanation

Let \ I = \int_{0}^{\infty} \frac{6}{e^{3x} + 6e^{2x} + 11e^{x} + 6} dx

I = \int_{0}^{\infty} \frac{6 \ dx}{(e^x + 1)(e^x + 2)(e^x + 3)} \ \text{(on factorising the Dr)}

Let \ \frac{6}{(e^x + 1)(e^x + 2)(e^x + 3)} = \frac{A}{e^x + 1} + \frac{B}{e^x + 2} + \frac{C}{e^x + 3}

On \ solving, \ we \ get \ A = 3, \ B = -6, \ C = 3

so \ I = \int_{0}^{\infty} \frac{3}{e^x + 1} dx - \int_{0}^{\infty} \frac{6}{e^x + 2} dx + \int_{0}^{\infty} \frac{3}{e^x + 3} dx

so \ I = \int_{0}^{\infty} \frac{3}{e^x + 1} dx - \int_{0}^{\infty} \frac{6}{e^x + 2} dx + \int_{0}^{\infty} \frac{3}{e^x + 3} dx

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