Question

Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.

Options

• `x + log [(x − 2)^5/(x − 1)^2] +C`

• `x + log [(x − 1)^2/(x − 2)^5] +C`

• `x − log [(x − 2)^5/(x − 1)^2] +C`

• None of these

Answer 1

Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is `underlinebb(x + log [(x − 2)^5/(x − 1)^2] +C)`.

Explanation:

Here, since the highest powers of x in the numerator and denominator are equal, and the coefficients of x² are also equal, therefore

`∫(x^2 + 1)/((x − 1)(x − 2)) ≡ 1 + A/(x − 1 + x − 2)`

On solving, we get A = −2, B = 5

Thus `∫ (x^2 + 1)/((x − 1)(x − 2)) ≡ 1 − 2/(x − 1) + 5/(x − 2)`

The above method is used to obtain the value of the constant corresponding to a non-repeated linear factor in the denominator.

Now, I =  `∫ (1 − 2/(x − 1) + 5/(x − 2))` dx

= x − 2 log (x − 1) + 5 log (x − 2) + C

= `x + log[(x − 2)^5/(x − 1)^2] + C`