Evaluate the Integral integral of xe^(-x^2) with respect to x

Problem

(∫_^)(x*e(−x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the exponent −x2 is proportional to the factor x outside the exponential.

2. Define the substitution variable u=−x2

3. Differentiate u with respect to x to find d(u)=−2*x*d(x)

4. Rearrange the differential to solve for the terms present in the integral, which gives x*d(x)=−1/2*d(u)

5. Substitute these expressions into the original integral to rewrite it in terms of u

(∫_^)(x*e(−x2)*d(x))=(∫_^)(eu*(−1/2)*d(u))

6. Factor out the constant −1/2 from the integral.

−1/2*(∫_^)(eu*d(u))

7. Integrate the exponential function, noting that (∫_^)(eu*d(u))=eu+C

−1/2*eu+C

8. Back-substitute the original expression for u to return to the variable x

−1/2*e(−x2)+C

Final Answer

(∫_^)(x*e(−x2)*d(x))=−1/2*e(−x2)+C

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