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प्रश्न
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
उत्तर
Let I = `int (2x)/((2 + x^2)(3 + x^2))*dx`
Put x2 = t
∴ 2x dx = dt
∴ I = `int (1)/((2 + t)(3 + t))*dt`
= `int ((3 + t) - (2 + t))/((2 + t)(3 + t))*dt`
= `int [1/(2 + t) - 1/(3 + t)]*dt`
= `int (1)/(2 + t)*dt - int (1)/(3 + t)*dt`
= log|2 + t| – log|3 + t| + c
= `log|(2 + t)/(3 + t)| + c`
= `log|(2 + x^2)/(3 + x^2)| + c`.
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