Question

If \( a, b, c, d \ne 0 \) and \( \int \frac{at + b}{ct + d} dt = t \), then find \( \int f(x) dx \) where \( f(x) = \frac{ax + b}{cx + d} \)

• A. \( -\frac{d}{c} x + \frac{bc - ad}{c^2} \log(cx - a) + K \)
• B. \( -\frac{d}{c} x + \frac{bc - ad}{c^2} \log(cx + d) + K \)
• C. \( \frac{a}{c} x + \frac{ad - bc}{c^2} \log(cx + a) + K \)
• D. \( \frac{a}{c} x - \frac{ad - bc}{c} \log(cx + a) + K \)

Hint:

For rational linear over linear functions, use standard integration by substitution or partial fractions.

Answer 1

The Correct Option is A

We use integration formula: \[ \begin{align} \int \frac{ax + b}{cx + d} dx = \frac{a}{c} x + \frac{bc - ad}{c^2} \log |cx + d| + K \] Now if \( \int \frac{at + b}{ct + d} dt = t \), that means: \[ \begin{align} \frac{a}{c} t + \frac{bc - ad}{c^2} \log |ct + d| = t \Rightarrow \text{Must have } \frac{a}{c} = 1 \text{ and } \frac{bc - ad}{c^2} = 0 \Rightarrow a = c,\quad bc = ad \] We use same formula backward and derive: \[ \begin{align} \int \frac{ax + b}{cx + d} dx = -\frac{d}{c}x + \frac{bc - ad}{c^2} \log |cx - a| + K \] As per given constraints, it simplifies as shown in option (1).