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Class 12 Integrals · Short Answer2 Marksindefinite_integration

Class 12 Maths · Integrals · Short Answer #141

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

✅ Step-by-Step Solution

Final Answer: \(\frac{e^{x}}{\left(x + 5\right)^{2}}\) + C
  1. By the product rule, d[e^x(x+5)^(−2)]/dx=e^x[(x+5)^(−2)−2(x+5)^(−3)].
  2. Combine the numerator to e^x[(x+5)−2]/(x+5)³=e^x(x+3)/(x+5)³.
  3. Therefore the required integral is \(\frac{e^{x}}{\left(x + 5\right)^{2}}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] By the product rule, d[e^x(x+5)^(−2)]/dx=e^x[(x+5)^(−2)−2(x+5)^(−3)].
  • [1 mark] Combine the numerator to e^x[(x+5)−2]/(x+5)³=e^x(x+3)/(x+5)³.
  • [0.5 marks] Therefore the required integral is \(\frac{e^{x}}{\left(x + 5\right)^{2}}\) + C.

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