Evaluate the integral. (Complete the square if necessary) `int (2x-5)/(x^2 + 2x + 2) dx`

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mathsworkmusic | (Level 2) Educator

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To evaluate this integral, first complete the square (as suggested) in the denominator of the integrand, giving

`I = int (2x-5)/((x+1)^2 + 1) quad dx`

Note that the derivative of the denominator is `2(x+1)`.

Importantly, recall that `int (g'(x))/g(x)" "dx = ln g(x)` ``````

We can then split the integral `I` into two components:

`I = int (2(x+1))/((x+1)^2 + 1)" "dx quad - quad 7 int 1/((x+1)^2 + 1)" "dx`

`= ln{(x+1)^2 + 1} quad - quad 7 int 1/((x+1)^2+1)" " dx`

The form of the integrand in the second component can be recognised as the derivative of `tan^(-1)(x)` but where `x` is replaced by `(x+1)` so that we have finally

`I = ln{(x+1)^2 + 1} - 7tan^-1(x+1) + c`

where `c` is a constant of integration