Integrals
Six special integrals cover every quadratic denominator:
| Integral | Result |
|---|---|
| \(\int\frac{dx}{x^2-a^2}\) | \(\frac1{2a}\log\left|\frac{x-a}{x+a}\right|+C\) |
| \(\int\frac{dx}{a^2-x^2}\) | \(\frac1{2a}\log\left|\frac{a+x}{a-x}\right|+C\) |
| \(\int\frac{dx}{x^2+a^2}\) | \(\frac1a\tan^{-1}\frac xa+C\) |
| \(\int\frac{dx}{\sqrt{x^2-a^2}}\) | \(\log\left|x+\sqrt{x^2-a^2}\right|+C\) |
| \(\int\frac{dx}{\sqrt{a^2-x^2}}\) | \(\sin^{-1}\frac xa+C\) |
| \(\int\frac{dx}{\sqrt{x^2+a^2}}\) | \(\log\left|x+\sqrt{x^2+a^2}\right|+C\) |
Completing the square. For \(\int\frac{dx}{ax^2+bx+c}\) or \(\int\frac{dx}{\sqrt{ax^2+bx+c}}\), write the quadratic as \(a[(x+h)^2\pm k^2]\) and use the table. Example: \(x^2+4x+13=(x+2)^2+9\), so \(\int\frac{dx}{x^2+4x+13}=\frac13\tan^{-1}\frac{x+2}{3}+C\).
Linear numerator. For \(\int\frac{px+q}{ax^2+bx+c}dx\), write \(px+q=A\frac{d}{dx}(ax^2+bx+c)+B\); the first part gives a logarithm and the second a standard form.
| Integral | Result |
|---|---|
| \(\int\frac{dx}{x^2-a^2}\) | \(\frac1{2a}\log\left|\frac{x-a}{x+a}\right|+C\) |
| \(\int\frac{dx}{a^2-x^2}\) | \(\frac1{2a}\log\left|\frac{a+x}{a-x}\right|+C\) |
| \(\int\frac{dx}{x^2+a^2}\) | \(\frac1a\tan^{-1}\frac xa+C\) |
| \(\int\frac{dx}{\sqrt{x^2-a^2}}\) | \(\log\left|x+\sqrt{x^2-a^2}\right|+C\) |
| \(\int\frac{dx}{\sqrt{a^2-x^2}}\) | \(\sin^{-1}\frac xa+C\) |
| \(\int\frac{dx}{\sqrt{x^2+a^2}}\) | \(\log\left|x+\sqrt{x^2+a^2}\right|+C\) |
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Start PracticeReduce to one of the six standard forms by completing the square; split linear numerators.
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