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secondary 4 | A Maths
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I tried substitution and integration by parts, but can't seem to work it out.
Date Posted: 3 years ago
Views: 466
Choose e⁻ˣ to integrate first.
∫ x² e⁻ˣ dx
= x² (-e⁻ˣ) - ∫ (2x(-e⁻ˣ)) dx
= -x²e⁻ˣ + 2 ∫ xe⁻ˣ dx
Here, choose the same term to integrate.
= -x²e⁻ˣ + 2 (x(-e⁻ˣ) - ∫( (1)(-e⁻ˣ) dx )
= -x²e⁻ˣ + 2 (-xe⁻ˣ + ∫e⁻ˣ dx)
= -x²e⁻ˣ + 2 (-xe⁻ˣ - e⁻ˣ )
= -x²e⁻ˣ - 2xe⁻ˣ - 2e⁻ˣ
= -e⁻ˣ (x² + 2x + 2) + c
Solving for the definite integral,
∫¹₀ x² e⁻ˣ dx
= [-e⁻ˣ (x² + 2x + 2)]¹₀
= -e⁻¹(1² + 2(1) + 2) - (-e⁻⁰ (0² + 2(0) + 2) )
= -5e⁻¹ + 2e⁻⁰
= 2/e⁰ - 5/e
= 2/1 - 5/e
= 2 - 5/e
∫ x² e⁻ˣ dx
= x² (-e⁻ˣ) - ∫ (2x(-e⁻ˣ)) dx
= -x²e⁻ˣ + 2 ∫ xe⁻ˣ dx
Here, choose the same term to integrate.
= -x²e⁻ˣ + 2 (x(-e⁻ˣ) - ∫( (1)(-e⁻ˣ) dx )
= -x²e⁻ˣ + 2 (-xe⁻ˣ + ∫e⁻ˣ dx)
= -x²e⁻ˣ + 2 (-xe⁻ˣ - e⁻ˣ )
= -x²e⁻ˣ - 2xe⁻ˣ - 2e⁻ˣ
= -e⁻ˣ (x² + 2x + 2) + c
Solving for the definite integral,
∫¹₀ x² e⁻ˣ dx
= [-e⁻ˣ (x² + 2x + 2)]¹₀
= -e⁻¹(1² + 2(1) + 2) - (-e⁻⁰ (0² + 2(0) + 2) )
= -5e⁻¹ + 2e⁻⁰
= 2/e⁰ - 5/e
= 2/1 - 5/e
= 2 - 5/e
Thanks...
No problem
See 1 Answer
You'll need to perform integration by parts more than once. Check the comments for the working.
Thanks...
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