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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #83

Evaluate \(\int \frac{\left(2 x - 5\right) e^{2 x}}{\left(2 x - 3\right)^{3}}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\frac{e^{2 x}}{2 \left(2 x - 3\right)^{2}}\) + C
  1. Let u=2x−3, so dx=du/2 and e^(2x)=e³e^u.
  2. The integral is (e³/2)∫e^u(u^(−2)−2u^(−3))du.
  3. The integrand in u is d(e^u/u²)/du; integrate and replace u=2x−3.
  4. Therefore the required integral is \(\frac{e^{2 x}}{2 \left(2 x - 3\right)^{2}}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Let u=2x−3, so dx=du/2 and e^(2x)=e³e^u.
  • [1 mark] The integral is (e³/2)∫e^u(u^(−2)−2u^(−3))du.
  • [1 mark] The integrand in u is d(e^u/u²)/du; integrate and replace u=2x−3.
  • [0.5 marks] Therefore the required integral is \(\frac{e^{2 x}}{2 \left(2 x - 3\right)^{2}}\) + C.

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