Integration by Substitution/Examples/(2x + 3) by Root of x^2 + 3x + 2

Example of Use of Integration by Substitution

$\ds \int \paren {2 x + 3} \sqrt {x^2 + 3 x + 2} \rd x = \dfrac 2 3 {\paren {\sqrt {x^2 + 3 x + 2} }^3} + C$

Proof

\(\ds u\)

\(=\)

\(\ds x^2 + 3 x + 2\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d u} {\d x}\)

\(=\)

\(\ds 2 x + 3\)

Power Rule for Derivatives

\(\ds \leadsto \ \ \)

\(\ds \int \paren {2 x + 3} \sqrt {x^2 + 3 x + 2} \rd x\)

\(=\)

\(\ds \int \sqrt u \rd u\)

Primitive of Composite Function: Corollary

\(\ds \)

\(=\)

\(\ds \dfrac {2 u^{3/2} } 3 + C\)

Primitive of Square Root

\(\ds \)

\(=\)

\(\ds \dfrac 2 3 {\paren {\sqrt {x^2 + 3 x + 2} }^3} + C\)

substituting for $u$ and simplifying

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $6$.