Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #156

Evaluate \(\int \frac{1 - x^{2}}{x \left(1 - 2 x\right)}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\frac{x}{2} + \log{\left(\left|{x}\right| \right)} - \frac{3 \log{\left(\left|{2 x - 1}\right| \right)}}{4}\) + C

Polynomial division and partial fractions give (1−x²)/[x(1−2x)]=1/2+1/x+3/[2(1−2x)].
Integrate the constant and reciprocal x directly. For the last term let u=1−2x, du=−2dx.
The last primitive is −(3/4)ln|1−2x|, which is the same as −(3/4)ln|2x−1|.
Therefore the required integral is \(\frac{x}{2} + \log{\left(\left|{x}\right| \right)} - \frac{3 \log{\left(\left|{2 x - 1}\right| \right)}}{4}\) + C.

[0.5 marks] Polynomial division and partial fractions give (1−x²)/[x(1−2x)]=1/2+1/x+3/[2(1−2x)].
[1 mark] Integrate the constant and reciprocal x directly. For the last term let u=1−2x, du=−2dx.
[1 mark] The last primitive is −(3/4)ln|1−2x|, which is the same as −(3/4)ln|2x−1|.
[0.5 marks] Therefore the required integral is \(\frac{x}{2} + \log{\left(\left|{x}\right| \right)} - \frac{3 \log{\left(\left|{2 x - 1}\right| \right)}}{4}\) + C.

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