First slide

Methods of integration

Question

Statement 1: For $- 1 < a < 4$
$\int \frac{d x}{x^{2} + 2 (a - 1) x + a + 5} = λ \log | g (x) | + C,$ where $λ$ and $C$
are constants
Statement 2: For $- 1 < a < 4, f (x) = \frac{1}{x^{2} + 2 (a - 1) x + a + 5}$ is
a continuous function.

Moderate

A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for
STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

The discriminant of $x^{2} + 2 (a - 1) x + a + 5 = 0$
is $4 (a - 1)^{2} - 4 (a + 5) = 4 (a^{2} - 3 a - 4) < 0$ hence the integral
$\int \frac{d x}{x^{2} + 2 (a - 1) x + a + 5} = λ \tan^{- 1} g (x) + C$
Since the denominator is not zero for any value of $x$ so $f$ is a
continuous function.

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