Question

$\int e^x(x^3-2x^2+3x-4)dx =$

• A. $-e^x(x^3-x^2+5x-1)+c$
• B. $e^x(x^3-x^2+5x-1)+c$
• C. $e^x(x^3+x^2+5x+1)+c$
• D. $-e^x(x^3+x^2+5x+1)+c$
• E. None of these

Hint:

For integrals of the form $\int e^{ax}P(x)dx$, you can use the tabular integration method (a shortcut for repeated integration by parts) or use the formula $\int e^x P(x)dx = e^x(P(x)-P'(x)+P''(x)-...)+C$. In this case, $P(x)=x^3-2x^2+3x-4$, $P'(x)=3x^2-4x+3$, $P''(x)=6x-4$, $P'''(x)=6$. The integral is $e^x((x^3-2x^2+3x-4)-(3x^2-4x+3)+(6x-4)-6)+C = e^x(x^3-5x^2+13x-17)+C$.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
This integral is of the form $\int e^x P(x) dx$, where $P(x)$ is a polynomial. A standard method for this is to use the property $\int e^x[f(x)+f'(x)]dx = e^x f(x) + C$. We need to express the polynomial $P(x)$ as the sum of some function $f(x)$ and its derivative $f'(x)$.
Step 2: Key Formula or Approach
1. Let the integrand be $e^x(f(x)+f'(x))$. We need to find a function $f(x)$ such that $f(x)+f'(x) = x^3-2x^2+3x-4$. 2. Since the derivative reduces the degree of a polynomial by one, $f(x)$ must be a cubic polynomial of the form $f(x) = Ax^3+Bx^2+Cx+D$. 3. We find $f'(x)$, add it to $f(x)$, and compare the coefficients with the given polynomial to solve for A, B, C, and D. 4. The result of the integration will be $e^x f(x) + C$.
Step 3: Detailed Explanation
Let $f(x) = Ax^3+Bx^2+Cx+D$. Then its derivative is $f'(x) = 3Ax^2+2Bx+C$. We are given that $f(x)+f'(x) = x^3-2x^2+3x-4$. \[ (Ax^3+Bx^2+Cx+D) + (3Ax^2+2Bx+C) = x^3-2x^2+3x-4 \] Group the terms by powers of $x$: \[ Ax^3 + (B+3A)x^2 + (C+2B)x + (D+C) = x^3-2x^2+3x-4 \] Now, equate the coefficients of corresponding powers of $x$: - Coefficient of $x^3$: $A = 1$. - Coefficient of $x^2$: $B+3A = -2 \implies B+3(1) = -2 \implies B = -5$. - Coefficient of $x$: $C+2B = 3 \implies C+2(-5) = 3 \implies C-10=3 \implies C = 13$. - Constant term: $D+C = -4 \implies D+13 = -4 \implies D = -17$. So, the function is $f(x) = x^3-5x^2+13x-17$. The integral is therefore: \[ \int e^x(x^3-2x^2+3x-4)dx = e^x f(x) + C = e^x(x^3-5x^2+13x-17) + C \] This result does not match any of the given options. This indicates that the problem statement (the polynomial in the integrand) or the options provided are incorrect. For example, if the integrand had been $e^x(x^3+4x^2+7x+6)$, then by a similar procedure we would find $f(x)=x^3+x^2+5x+1$, and the answer would be $e^x(x^3+x^2+5x+1)+C$, matching option (C). Step 4: Final Answer
Based on a rigorous calculation, the correct integral is $e^x(x^3-5x^2+13x-17)+C$, which is not among the options. The question is flawed.