Primitive of Reciprocal of a x + b cubed/Proof 1
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Theorem
- $\ds \int \frac {\d x} {\paren {a x + b}^3} = -\frac 1 {2 a \paren {a x + b}^2} + C$
Proof
Let $u = a x + b$.
Then:
\(\ds \int \frac {\rd x} {\paren {a x + b}^3}\) | \(=\) | \(\ds \frac 1 a \int \frac {\rd u} {u^3}\) | Primitive of Function of $a x + b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac {-1} {2 u^2} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {2 a \paren {a x + b}^2} + C\) | substituting for $u$ |
$\blacksquare$