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Evaluate: integral of dx / ((x-1)^(3/4) * (x+2)^(5/4)).

1. (3/4) * ((x+2)/(x-1))^(1/4) + C
2. (3/4) * ((x+2)/(x-1))^(5/4) + C
3. (4/3) * ((x-1)/(x+2))^(5/4) + C
4. (4/3) * ((x-1)/(x+2))^(1/4) + C

Correct answer: (4/3) * ((x-1)/(x+2))^(1/4) + C

Solution

Write the integral as integral dx / ((x-1)^(3/4)(x+2)^(5/4)). Let t=(x-1)/(x+2), dt=3/(x+2)² dx. Rewrite integrand: 1/((x-1)^(3/4)(x+2)^(5/4)) = (1/(x+2)²) * (x+2)^(3/4)/(x-1)^(3/4) = (1/(x+2)²) * (1/t)^(3/4)... better: = (x+2)^(-2) * ((x-1)/(x+2))^(-3/4) = (x+2)^(-2) * t^(-3/4). So integral = integral t^(-3/4) * (x+2)^(-2) dx = (1/3) integral t^(-3/4) dt = (1/3)*(t^(1/4)/(1/4)) + C = (4/3)*t^(1/4)+C = (4/3)*((x-1)/(x+2))^(1/4)+C.

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