Primitive of x over Root of a x squared plus b x plus c/Examples/x over Root of x^2 + 4 x + 5

Examples of Use of Primitive of $\dfrac x {\sqrt {a x^2 + b x + c} }$

$\ds \int \dfrac {x \rd x} {\sqrt {x^2 + 4 x + 5} } = \sqrt {x^2 + 4 x + 5} - 2 \map \ln {x + 2 + \sqrt {x^2 + 4 x + 5} } + C$

Proof

From Primitive of $\dfrac x {\sqrt {a x^2 + b x + c} }$ with:

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

Then for $x \in \R$ such that $a x^2 + b x + c > 0$:

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

\(\ds \int \dfrac {x \rd x} {\sqrt {x^2 + 4 x + 5} }\)

\(=\)

\(\ds \frac {\sqrt {x^2 + 4 x + 5} } 1 - \frac 4 {2 \times 1} \int \frac {\d x} {\sqrt {x^2 + 4 x + 5} } + C\)

substituting $a \gets 1$, $b \gets 4$, $c \gets 5$

\(\ds \)

\(=\)

\(\ds \sqrt {x^2 + 4 x + 5} - 2 \int \frac {\d x} {\sqrt {x^2 + 4 x + 5} } + C\)

simplifying

\(\ds \)

\(=\)

\(\ds \sqrt {x^2 + 4 x + 5} - 2 \map \ln {x + 2 + \sqrt {x^2 + 4 x + 5} } + C\)

Primitive of $\dfrac 1 {\sqrt {x^2 + 4 x + 5} }$

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $18$.