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

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

$\ds \int \dfrac {x \rd x} {x^2 + 4 x + 5} = \frac 1 2 \map \ln {x^2 + 4 x + 5} - 2 \map \arctan {x + 2} + C$

Proof

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

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

Then:

$\ds \int \frac {x \rd x} {a x^2 + b x + c} = \frac 1 {2 a} \ln \size {a x^2 + b x + c} - \frac b {2 a} \int \frac {\d x} {a x^2 + b x + c}$

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

\(=\)

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

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

\(\ds \)

\(=\)

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

simplifying: $x^2 + 4 x + 5 > 0$ for all $x \in \R$

\(\ds \)

\(=\)

\(\ds \frac 1 2 \map \ln {x^2 + 4 x + 5} - 2 \map \arctan {x + 2} + C\)

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

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $17$.