Primitive of Root of a x squared plus b x plus c

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Theorem

Let $a \in \R_{\ne 0}$.

Then:

$\ds \int \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }$


Proof

Let:

\(\ds z\) \(=\) \(\ds \paren {2 a x + b}^2\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds 4 a \paren {2 a x + b}\) Derivative of Power and Chain Rule for Derivatives
\(\ds \leadsto \ \ \) \(\ds \frac {\d z} {\d x}\) \(=\) \(\ds 4 a \sqrt z\)


Suppose $a > 0$.

Then we have:

\(\ds \) \(\) \(\ds \int \sqrt {a x^2 + b x + c} \rd x\)
\(\ds \) \(=\) \(\ds \int \sqrt {\frac {\paren {2 a x + b}^2 + \paren {4 a c - b^2} } {4 a} } \rd x\) Completing the Square
\(\ds \) \(=\) \(\ds \int \frac {\sqrt {z + \paren {4 a c - b^2} } \rd z} {\paren {2 \sqrt a} \paren {4 a \sqrt z} }\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac 1 {8 a \sqrt a} \int \frac {\sqrt {z + \paren {4 a c - b^2} } \rd z} {\sqrt z}\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \frac 1 {8 a \sqrt a} \paren {\sqrt z \sqrt {z + \paren {4 a c - b^2} } + \frac {4 a c - b^2} 2 \int \frac {\d z} {\sqrt z \sqrt {z + \paren {4 a c - b^2} } } }\) Primitive of $\dfrac {\sqrt{p x + q} } {\sqrt{a x + b} }$
\(\ds \) \(=\) \(\ds \frac {\sqrt z \sqrt {z + \paren {4 a c - b^2} } } {8 a \sqrt a} + \frac {4 a c - b^2} {16 a \sqrt a} \int \frac {\d z} {\sqrt z \sqrt {z + \paren {4 a c - b^2} } }\) multiplying out
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b} \sqrt {\paren {2 a x + b}^2 + \paren {4 a c - b^2} } } {8 a \sqrt a} + \frac {4 a c - b^2} {16 a \sqrt a} \int \frac {4 a \rd x} {\sqrt {\paren {2 a x + b}^2 + \paren {4 a c - b^2} } }\) substituting back for $z$ and $\d z$
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b} \paren {2 \sqrt a \sqrt {a x^2 + b x + c} } } {8 a \sqrt a} + \frac {4 a c - b^2} {16 a \sqrt a} \int \frac {4 a \rd x} {2 \sqrt a \sqrt {a x^2 + b x + c} }\) substituting back for $\sqrt {a x^2 + b x + c}$
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) simplifying


When $a < 0$, the above does not work, as we cannot take the square root of a negative number.

Hence in this case:

\(\ds \) \(\) \(\ds \int \sqrt {a x^2 + b x + c} \rd x\)
\(\ds \) \(=\) \(\ds \int \sqrt {\frac {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } {4 a} } \rd x\) Completing the Square
\(\ds \) \(=\) \(\ds \int \sqrt {\frac {\paren {b^2 - 4 a c} - \paren {2 a x + b}^2} {-4 a} } \rd x\) Completing the Square
\(\ds \) \(=\) \(\ds \int \frac {\sqrt {\paren {b^2 - 4 a c} - z} \rd z} {\paren {2 \sqrt {-a} } \paren {4 a \sqrt z} }\) Integration by Substitution
\(\ds \) \(=\) \(\ds \frac 1 {8 a \sqrt {-a} } \int \frac {\sqrt {-z + \paren {b^2 - 4 a c} } \rd z} {\sqrt z}\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \frac 1 {8 a \sqrt {-a} } \paren {\sqrt z \sqrt {-z + \paren {b^2 - 4 a c} } + \frac {b^2 - 4 a c} 2 \int \frac {\d z} {\sqrt z \sqrt {-z + \paren {b^2 - 4 a c} } } }\) Primitive of $\dfrac {\sqrt{p x + q} } {\sqrt{a x + b} }$
\(\ds \) \(=\) \(\ds \frac {\sqrt z \sqrt {\paren {b^2 - 4 a c} - z} } {8 a \sqrt {-a} } + \frac {b^2 - 4 a c} {16 a \sqrt {-a} } \int \frac {\d z} {\sqrt z \sqrt {\paren {b^2 - 4 a c} - z} }\) multiplying out
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b} \sqrt {\paren {b^2 - 4 a c} - \paren {2 a x + b}^2} } {8 a \sqrt {-a} } + \frac {b^2 - 4 a c} {16 a \sqrt {-a} } \int \frac {4 a \rd x} {\sqrt {\paren {b^2 - 4 a c} - \paren {2 a x + b}^2} }\) substituting back for $z$ and $\d z$
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b} \paren {2 \sqrt {-a} \sqrt {a x^2 + b x + c} } } {8 a \sqrt {-a} } + \frac {b^2 - 4 a c} {16 a \sqrt {-a} } \int \frac {4 a \rd x} {2 \sqrt {-a} \sqrt {a x^2 + b x + c} }\) substituting back for $\sqrt {a x^2 + b x + c}$
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} + \frac {b^2 - 4 a c} {-8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) simplifying
\(\ds \) \(=\) \(\ds \frac {\paren {2 a x + b} \sqrt {a x^2 + b x + c} } {4 a} +\frac {4 a c - b^2} {8 a} \int \frac {\d x} {\sqrt {a x^2 + b x + c} }\) arranging it into its final form

$\blacksquare$


Sources

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