સંમેય વિધેયનું સંકલન કરો: $\frac{2x-3}{(x^2-1)(2x+3)}$

(N/A) આપણી પાસે $\frac{2x-3}{(x^2-1)(2x+3)} = \frac{2x-3}{(x+1)(x-1)(2x+3)}$ છે.
ધારો કે $\frac{2x-3}{(x+1)(x-1)(2x+3)} = \frac{A}{x+1} + \frac{B}{x-1} + \frac{C}{2x+3}$.
તેથી $2x-3 = A(x-1)(2x+3) + B(x+1)(2x+3) + C(x+1)(x-1)$.
$x=1$ લેતા: $2(1)-3 = B(1+1)(2(1)+3) \Rightarrow -1 = B(2)(5) \Rightarrow B = -\frac{1}{10}$.
$x=-1$ લેતા: $2(-1)-3 = A(-1-1)(2(-1)+3) \Rightarrow -5 = A(-2)(1) \Rightarrow A = \frac{5}{2}$.
$x=-\frac{3}{2}$ લેતા: $2(-\frac{3}{2})-3 = C(-\frac{3}{2}+1)(-\frac{3}{2}-1) \Rightarrow -6 = C(-\frac{1}{2})(-\frac{5}{2}) \Rightarrow -6 = C(\frac{5}{4}) \Rightarrow C = -\frac{24}{5}$.
આમ,$\int \frac{2x-3}{(x^2-1)(2x+3)} dx = \int \left( \frac{5}{2(x+1)} - \frac{1}{10(x-1)} - \frac{24}{5(2x+3)} \right) dx$.
$= \frac{5}{2} \log|x+1| - \frac{1}{10} \log|x-1| - \frac{24}{5} \cdot \frac{1}{2} \log|2x+3| + K$.
$= \frac{5}{2} \log|x+1| - \frac{1}{10} \log|x-1| - \frac{12}{5} \log|2x+3| + K$,જ્યાં $K$ એ સંકલનનો અચળાંક છે.