This page contains solutions to the following problems

$$\int \! \frac{1}{x \left( M – x \right) } \, dx$$
$$\int \! \frac{1}{ \left( ax + b \right) \left( cx + d \right) } \, dx$$
$$\int \! \frac{ \left( kx + l \right) }{ \left( ax + b \right) \left( cx + d \right) } \ dx$$

\( \displaystyle \int \! \frac{1}{x \left( M – x \right) } \ dx \)

Begin by decomposing the integrand.

$$ \frac{1}{x \left( M – x \right) } = \frac{A}{ x } + \frac{B}{ M – x } $$

Multiply through by \( x \left( M – x \right) \)

$$ 1 = A \left( M – x \right) + B x $$

If \( x = M \) then \( B = \frac{1}{ M } \)

If \( x = 0 \) then \( A = \frac{1}{ M } \)

therefore,

$$ \int \! \frac{1}{x \left( M – x \right) } \, dx = \frac{1}{M} \int \! \frac{1}{ x } \, dx + \frac{1}{M} \int \! \frac{1}{ M – x } \, dx $$
$$ \int \! \frac{1}{x \left( M – x \right) } = \frac{1}{M} \ln \left| x \right| – \frac{1}{M} \ln \left| M – x \right| + C = \frac{1}{M} \ln \left| \frac{x}{M – x} \right| + C$$

\( \displaystyle \int \! \frac{1}{ \left( ax + b \right) \left( cx + d \right) } \, dx \)

Decompose the integrand

$$ \frac{1}{ \left( ax + b \right) \left( cx + d \right) } = \frac{A}{ \left( ax + b \right)} \frac{B}{ \left( cx + d \right) }$$

Multiply through by \( \left( ax + b \right) \left( cx + d \right) \)

$$ 1 = A \left( cx + d \right) + B \left( ax + b \right) $$

If \( x = – \frac{d}{c} \), then \( 1 = B \left( a \left( -\frac{d}{c} \right) + b \right) \) yielding

$$ B = \frac{c}{ bc – ad } $$

Simimlarly,

If \( x = – \frac{b}{a} \), then \( 1 = A \left( c \left( -\frac{b}{a} \right) + d \right) \) and

$$ A = \frac{a}{ ad – bc } $$

substituting

$$ \int \! \frac{1}{ \left( ax + b \right) \left( cx + d \right) } \, dx = \frac{a}{ ad – bc } \int \! \frac{1}{ \left( ax + b \right) } \, dx + \frac{c}{ bc – ad } \int \! \frac{1}{ \left( cx + d \right) } \, dx $$

\( p = a x + b \ \ \ \ \ \ dp = a \ dx \)

\( q = c x + d \ \ \ \ \ \ dq = c \ dx \)

$$ \int \! \frac{1}{ \left( ax + b \right) \left( cx + d \right) } \, dx = \frac{a}{ ad – bc } \frac{1}{a} \int \! \frac{1}{ p } \, dx + \frac{c}{ bc – ad } \frac{1}{c} \int \! \frac{1}{ q } \, dx $$
$$ \int \! \frac{1}{ \left( ax + b \right) \left( cx + d \right) } \ dx = \frac{1}{ ad – bc } \ln \left( ax + b \right) + \frac{1}{bc – ad} \ln \left( c x + d \right) + C = \frac{1}{ad-bc} \ln \left( \frac{ a x + b }{ c x + d } \right) + C $$

\( \displaystyle \int \! \frac{ \left( kx + l \right) }{ \left( ax + b \right) \left( cx + d \right) } \ dx \)

\( \frac{ k x + l }{ \left( ax + b \right) \left( cx + d \right) } = \frac{A}{ \left( ax + b \right)} \frac{B}{ \left( cx + d \right) }\)

\( k x + l = A \left( cx + d \right) + B \left( ax + b \right) \)

\( x = – \frac{d}{c} \)

\( k \left( – \frac{d}{c} \right) + l = B \left( a \left( -\frac{d}{c} \right) + b \right) \)

\( B = \frac{ c l – d k }{ bc – ad } \)

\( x = – \frac{b}{a} \)

\( k \left( – \frac{b}{a} \right) + l = A \left( c \left( -\frac{b}{a} \right) + d \right) \)

\( A = \frac{ a l – k b }{ ad – bc } \)

\( \int \! \frac{ \left( kx + l \right) }{ \left( ax + b \right) \left( cx + d \right) } \ dx = \frac{ a l – k b }{ ad – bc } \int \! \frac{1}{ a x + b } \ dx + \frac{ c l – d k }{ bc – ad } \int \! \frac{1}{ c x + d } \ dx \)

\( p = a x + b \ \ \ \ \ \ dp = a \ dx \)

\( q = c x + d \ \ \ \ \ \ dq = c \ dx \)

\( \int \! \frac{ k x + l }{ \left( ax + b \right) \left( cx + d \right) } \ dx = \frac{a l – k b}{ ad – bc } \frac{1}{a} \int \! \frac{1}{ p } \ dx + \frac{ c l – d k }{ bc – ad } \frac{1}{c} \int \! \frac{1}{ q } \ dx \)

\( \int \! \frac{ k x + l}{ \left( ax + b \right) \left( cx + d \right) } \ dx = \frac{a l – k b}{ a^2 d – abc } \ln \! \left( ax + b \right) + \frac{ c l – d k}{bc^2 – acd} \ln \! \left( c x + d \right) + C = \frac{ a l – k b}{ a^2 d – abc } \ln \! \left( ax + b \right) + \frac{ d k – c l }{acd – bc^2} \ln \! \left( c x + d \right) + C \)