Question

The value of $\int \frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)} dx$ is (where ' $C$ ' is the constant of integration)

(A) $x+3 \ln |x-4|-24 \ln |x-5|+30 \ln |x-6|+C$

(B) $x+3 \ln |x-4|-12 \ln |x-5|+30 \ln |x-6|+C$

(C) $x+3 \ln |x-4|-12 \ln |x-5|+36 \ln |x-6|+C$

(D) $x+12 \ln |x-4|-24 \ln |x-5|+30 \ln |x-6|+C$

✨ Free AI Tools for You

Powered by Chatterbot AI

Answer

The correct option is A:
$$ x + 3 \ln |x-4| - 24 \ln |x-5| + 30 \ln |x-6| + C $$

Given the integral: $$ I = \int \frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)} , dx $$

We can rewrite it as: $$ I = \int \left[ 1 + \frac{(x-1)(x-2)(x-3) - (x-4)(x-5)(x-6)}{(x-4)(x-5)(x-6)} \right] , dx $$

Simplifying the numerator $(x-1)(x-2)(x-3) - (x-4)(x-5)(x-6)$: $$ = \int \left[ 1 + \frac{3 \times 2 \times 1}{(x-4)(-1)(-2)} + \frac{4 \times 3 \times 2}{(1)(x-5)(-1)} + \frac{5 \times 4 \times 3}{(2)(1)(x-6)} \right] , dx $$

This simplifies to: $$ = \int \left[ 1 + \frac{3}{(x-4)} - \frac{24}{(x-5)} + \frac{30}{(x-6)} \right] , dx $$

Integrating term by term: $$ I = x + 3 \ln |x-4| - 24 \ln |x-5| + 30 \ln |x-6| + C $$

Thus, the value of the integral is: $$ x + 3 \ln |x-4| - 24 \ln |x-5| + 30 \ln |x-6| + C $$

So the correct option is A.


Was this helpful?

India's 1st AI Doubt Solver for CBSE, JEE, and NEET

Ask a Question for Free

and then it's just ₹212 a month