Primitive of Power of a x + b/Proof 3

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Theorem

$\ds \int \paren {a x + b}^n \rd x = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} + C$

where $n \ne 1$.


Proof

\(\ds \map {\dfrac \d {\d x} } {\frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} }\) \(=\) \(\ds \dfrac {\paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a} \map {\dfrac \d {\d x} } {a x + b}\) Power Rule for Derivatives, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac {a \paren {n + 1} \paren {a x + b}^n} {\paren {n + 1} a}\) Power Rule for Derivatives
\(\ds \) \(=\) \(\ds \paren {a x + b}^n\) simplifying

The result follows by definition of primitive.

$\blacksquare$


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