Primitive of Reciprocal of x by a x + b squared/Proof 1

Theorem

$\ds \int \frac {\d x} {x \paren {a x + b}^2} = \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C$

Proof

\(\ds \int \frac {\d x} {x \paren {a x + b}^2}\)

\(=\)

\(\ds \int \paren {\frac 1 {b^2 x} - \frac a {b^2 \paren {a x + b} } - \frac a {b \paren {a x + b}^2} } \rd x\)

Partial Fraction Expansion

\(\ds \)

\(=\)

\(\ds \frac 1 {b^2} \int \frac {\d x} x - \frac a {b^2} \int \frac {\d x} {a x + b} - \frac a b \int \frac {\d x} {\paren {a x + b}^2}\)

Linear Combination of Primitives

\(\ds \)

\(=\)

\(\ds \frac 1 {b^2} \ln \size x - \frac a {b^2} \int \frac {\d x} {a x + b} - \frac a b \int \frac {\d x} {\paren {a x + b}^2} + C\)

Primitive of Reciprocal

\(\ds \)

\(=\)

\(\ds \frac 1 {b^2} \ln \size x - \frac a {b^2} \ln \size {a x + b} - \frac a b \int \frac {\d x} {\paren {a x + b}^2} + C\)

Primitive of $\dfrac 1 {a x + b}$

\(\ds \)

\(=\)

\(\ds \frac 1 {b^2} \ln \size x - \frac a {b^2} \ln \size {a x + b} - \frac a b \frac {-1} {a \paren {a x + b} } + C\)

Primitive of $\dfrac 1 {\paren {a x + b}^2}$

\(\ds \)

\(=\)

\(\ds \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C\)

Difference of Logarithms and rearranging

$\blacksquare$