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