Question:
If \(\int x^{3}e^{x}dx=e^{x}(px^{3}+qx^{2}+rx+s)+c\), then the value of \(p+q+r+s\) is
If \(\int x^{3}e^{x}dx=e^{x}(px^{3}+qx^{2}+rx+s)+c\), then the value of \(p+q+r+s\) is
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Apply integration by parts repeatedly to get the polynomial multiplying e to the x.
Updated On: Nov 12, 2026
- \(0\)
- \(-2\)
- \(6\)
- \(-4\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
Use \(\int x^{n}e^{x}dx=x^{n}e^{x}-n\int x^{n-1}e^{x}dx\) repeatedly.
Step 2: Apply repeatedly
\(\int x^{3}e^{x}dx=e^{x}(x^{3}-3x^{2}+6x-6)+c\).
Step 3: Find the sum
Comparing, \(p=1,\ q=-3,\ r=6,\ s=-6\), so \(p+q+r+s=1-3+6-6=-2\).
Final Answer:
The sum is \(-2\). \[ \boxed{\text{(B) }-2} \]
Use \(\int x^{n}e^{x}dx=x^{n}e^{x}-n\int x^{n-1}e^{x}dx\) repeatedly.
Step 2: Apply repeatedly
\(\int x^{3}e^{x}dx=e^{x}(x^{3}-3x^{2}+6x-6)+c\).
Step 3: Find the sum
Comparing, \(p=1,\ q=-3,\ r=6,\ s=-6\), so \(p+q+r+s=1-3+6-6=-2\).
Final Answer:
The sum is \(-2\). \[ \boxed{\text{(B) }-2} \]
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