NCERT Class 12 · Ch 7 Exercise 7.6 · Q3
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 3
Evaluate \(\int x^{2} e^{x}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(\left(x^{2} - 2 x + 2\right) e^{x}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Choose u=\(x^{2}\) and v=\(e^{x}\), so dv=\(e^{x}\)dx and du=\(2 x\)dx.
- Apply ∫u dv=uv−∫v du. The remaining integrand v·u′ is \(2 x e^{x}\). Integrate it and subtract from uv.
- An antiderivative is F(x)=\(\left(x^{2} - 2 x + 2\right) e^{x}\).
- Differentiate this expression to check that F′(x)=\(x^{2} e^{x}\). Add an arbitrary constant C on each domain component.
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