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

Theorem

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

Proof

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

\(=\)

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

Partial Fraction Expansion

\(\ds \)

\(=\)

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

Linear Combination of Primitives

\(\ds \)

\(=\)

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

Primitive of Power

\(\ds \)

\(=\)

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

Primitive of Reciprocal

\(\ds \)

\(=\)

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

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

\(\ds \)

\(=\)

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

Difference of Logarithms

$\blacksquare$