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Q.
The value of definite integral $\int\limits_0^1 \frac{x^2+5}{(x+1)^2(x-2)} d x$ is equal to
Integrals
Solution:
$I=\int\limits_0^1 \frac{\left(x^2+2 x+1\right)-2(x-2)}{(x+1)^2(x-2)} d x$
$\left.\left. I =\int\limits_0^1 \frac{ dx }{ x -2}-2 \int_0^1 \frac{ dx }{( x +1)^2} \Rightarrow I =\ln | x -2|\right]_0^1+\frac{2}{ x +1}\right]_0^1=(0-\ln 2)+(1-2)$
$I =-(1+\ln 2) .$