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.
$\blacksquare$
Sources
- 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Three rules for integration: $\text {III}$