Primitive of x squared over a x + b squared/Proof 2

Theorem

$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2} = \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C$

Proof

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

\(=\)

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

multiplying top and bottom by $a$

\(\ds \)

\(=\)

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

adding and subtracting $b x$

\(\ds \)

\(=\)

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

Linear Combination of Primitives

\(\ds \)

\(=\)

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

simplification

\(\ds \)

\(=\)

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

Primitive of Reciprocal of $\dfrac x {\paren {a x + b} }$

\(\ds \)

\(=\)

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

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

\(\ds \)

\(=\)

\(\ds \frac x {a^2} - \frac b {a^3} \ln \size {a x + b} - \frac {b^2} {a^3 \paren {a x + b} } - \frac b {a^3} \ln \size {a x + b} + C\)

multiplying out

\(\ds \)

\(=\)

\(\ds \frac x {a^2} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C\)

gathering terms

\(\ds \)

\(=\)

\(\ds \frac {a x} {a^3} + \frac b {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C\)

where $\dfrac b {a^3}$ is subsumed into $C$

\(\ds \)

\(=\)

\(\ds \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C\)

simplification

$\blacksquare$