Evaluate the following integrals: 1 1+x-x^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{1}{1+\text{x}-\text{x}^2}\text{dx}$

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Answer

$\int\frac{\text{dx}}{1+\text{x}-\text{x}^2}$
$=\int\frac{-\text{dx}}{\text{x}^2-\text{x}-1}$
$=\int\frac{-\text{dx}}{\text{x}^2-\text{x}+\frac{1}{4}-\frac{1}{4}-1}$
$=\int\frac{-\text{dx}}{\big(\text{x}-\frac{1}{2}\big)^2-\frac{5}{4}}$
$=\int\frac{\text{dx}}{\frac{5}{4}-\big(\text{x}-\frac{1}{2}\big)^2}$

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