2x-3/(x²-1)(2x+3) - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.5
Integrate the rational functions in Exercises 111 to 131.

Question : 120 of 271

Marks: +1, -0

\frac{2x-3}{(x^2-1)(2x+3)}

Solution:

Let

I=\int\limits \frac{2x-3}{(x^2-1)(2x+3)}dx

Let

\frac{2x-3}{(x^2-1)(2x+3)}

=\frac{2x-3}{(x-1)(x+1)(2x+3)}

=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{2x+3}

⇒ 2x – 3 = A(x + 1)(2x + 3) + B(x – 1)(2x + 3) + C(x – 1)(x + 1) ...(i)

Putting x = 1 in (i), we get

2(1) – 3 = A(1 + 1) (2 + 3) ⇒ – 1 = A(2) (5) ⇒ = A

=-\frac{1}{10}

Putting x = –1 in (i), we get

– 2 – 3 = B(–1 – 1) (–2 + 3) ⇒ – 5 = B(–2) (1) ⇒ B

=\frac{5}{2}

Putting

x=-\frac{3}{2}

in (i), we get

-3-3=C\left(-\frac{3}{2}-1\right)\left(-\frac{3}{2}+1\right)

\Rightarrow -6=C\left(-\frac{5}{2}\right)\left(-\frac{1}{2}\right)

\Rightarrow C=-6\times\frac{4}{5}=-\frac{24}{5}

\frac{2x-3}{(x^2-1)(2x+3)} =-\frac{1}{10(x-1)}+\frac{5}{2(x+1)}-\frac{24}{5(2x+3)}

\therefore \int\limits \frac{2x-3}{(x^2-1)(2x+3)}dx =-\frac{1}{10}\int\limits \frac{dx}{x-1}+\frac{5}{2}\int\limits \frac{dx}{x+1}-\frac{24}{5}\int\limits \frac{dx}{2x+3}

=-\frac{1}{10}\log\left|x-1\right|+\frac{5}{2}\log\left|x-1\right|-\frac{12}{5}\log\left|2x+3\right|+C

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