I know how to integrate $\int \frac{ax±b}{cx±d} \, dx$ where $a, b, c, d$ are real numbers, but for such an easy looking integral it takes me too much time, so how should I approach this problem to get the result faster?
Let's say we have this integral $\int \frac{2x-3}{5x-2} \, dx$. I would substitute $t=5x-2$, $dt=5dx$ and after that it's easy but takes too much time. I would be happy for any advice.
Thanks.
On
$$\int\frac{ax\pm b}{cx\pm d}dx=\int\frac{\frac ac x\pm \frac bc}{x\pm \frac dc}dx=\int\frac{\frac ac(x\pm \frac dc)\mp\frac{ad}{c^2}\pm \frac bc}{x\pm \frac dc}dx=\frac ac x+\left(\mp\frac{ad}{c^2}\pm \frac bc\right)\int\frac 1{x\pm \frac dc}dx$$
On
Hint:
Canonical form of a homographic function: $$\frac{2x-3}{5x-2}=\frac{\frac25(5x-2)-\frac{11}5}{5x-2}=\frac 25-\frac{11}5\frac{1}{5x-2}=\frac 25-\frac{11}{25}\frac{1}{x-\frac25}$$
On
Any rational function, of which this is a particularly simple example, can be integrated using the method of partial fractions .
First, we can absorb the $\pm$ symbols into $b, d$, so we may as well take them both to be $+$.
In the generic case $a, c \neq 0$, se can pull off the leading coefficients andwrite $$\frac{a x + b}{c x + d} = \frac{a}{c} \cdot \frac{x + p}{x + q}$$ for $p = \frac{b}{a}, q = \frac{d}{c}$ and then rewrite $$\frac{x + p}{x + q} = \frac{x + q + (p - q)}{x + q} = 1 + (p - q) \cdot \frac{1}{x + q} .$$ Integrating, putting this all together, and making a slightly clever choice of constant that gives a cleaner expression gives: $$\color{#df0000}{\boxed{\int \frac{a x + b}{c x + d} \,dx = \frac{a}{c} x - \frac{a d - b c}{c^2} \log |c x + d| + C}}.$$
Of course, the cases with $a = 0$ and/or $c = 0$ immediately reduce to standard elementary integrals.