Genuine Genius Good Math Question?

By division, show that (x³/2x-1)=(x²/2)+(x/4)+1/8+(1/8(2x-1))...‡ and evaluate ∫x³ dx/2x-1.

Ans given:

(x³/6)+(x²/8)+(x/8)+(1/16)log(2x-1)

Update:

ignore the '...', technical problem

1 Answer

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  • Anonymous
    2 years ago
    Favorite Answer

    I shall do this the lazy way

    int (x^3)/(2x - 1) dx; Let u = 2x - 1; du = 2 dx; (u + 1)/2 = x

    1/2 * int (((u + 1)/2)^3)/u du;

    1/16 * int ((u + 1)^3)/u du; expand and divide

    1/16 * int ((u^3) + 3(u^2) + 3u + 1)/u du =

    1/16 * int ((u^2) + 3u + 3 + (1/u)) du =

    1/16 * [((u^3)/3) + (3(u^2)/2) + 3u + ln|u|] + K;

    1/16 * [(((2x - 1)^3)/3) + (3((2x - 1)^2)/2) + 3(2x - 1) + ln|2x - 1|] + K;

    (1/16)*ln|2x - 1| + ((x^3)/6) + ((x^2)/8) + (x/8) - 11/96 + K;

    C = K - (11/96);

    ((x^3)/6) + ((x^2)/8) + (x/8) + (1/16)*ln|2x - 1| + C

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