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 a multiple of the factor x outside the exponential.

2. Substitute u=x2 to simplify the integrand.

3. Differentiate u with respect to x to find d(u)=2*x*d(x) which implies x*d(x)=1/2*d(u)

4. Rewrite the integral in terms of u by substituting the expressions found in the previous steps.

(∫_^)(x*e(x2)*d(x))=(∫_^)(eu1/2*d(u))

5. Factor out the constant 1/2 from the integral.

1/2*(∫_^)(eu*d(u))

6. Integrate the exponential function eu which results in eu+C

1/2*eu+C

7. 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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