Primitive of p x + q over Root of a x squared plus 2 b x plus c/Examples/1 + x over Root 2 + 4 x - 3 x^2

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

$\ds \int \dfrac {1 + x} {\sqrt {2 + 4 x - 3 x^2} } \rd x = -\dfrac 1 3 \sqrt {2 + 4 x - 3 x^2} + \dfrac 5 {3 \sqrt 3} \map \arcsin {\dfrac {3 x - 2} {\sqrt {10} } } + C$

Proof

\(\ds \int \dfrac {1 + x} {\sqrt {2 + 4 x - 3 x^2} } \rd x\)

\(=\)

\(\ds -\dfrac 1 6 \int \dfrac {-6 - 6 x} {\sqrt {2 + 4 x - 3 x^2} } \rd x\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds -\dfrac 1 6 \paren {\int \dfrac {4 - 6 x} {\sqrt {2 + 4 x - 3 x^2} } \rd x + \int \dfrac {10 \rd x} {\sqrt {2 + 4 x - 3 x^2} } }\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds -\dfrac 1 6 \int \dfrac {4 - 6 x} {\sqrt {2 + 4 x - 3 x^2} } \rd x + \dfrac 5 3 \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} }\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds -\dfrac 1 6 \paren {2 \sqrt {2 + 4 x - 3 x^2} } + \dfrac 5 3 \int \dfrac {\d x} {\sqrt {2 + 4 x - 3 x^2} }\)

Square Root of Function under Derivative

\(\ds \)

\(=\)

\(\ds -\dfrac 1 3 \sqrt {2 + 4 x - 3 x^2} + \dfrac 5 3 \paren {\dfrac 1 {\sqrt 3} \map \arcsin {\dfrac {3 x - 2} {\sqrt {10} } } } + C\)

Primitive of $\dfrac 1 {\sqrt {2 + 4 x - 3 x^2} }$

\(\ds \)

\(=\)

\(\ds -\dfrac 1 3 \sqrt {2 + 4 x - 3 x^2} + \dfrac 5 {3 \sqrt 3} \map \arcsin {\dfrac {3 x - 2} {\sqrt {10} } } + C\)

simplifying

$\blacksquare$

Sources

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