Evaluate the Integral

Problem

(∫_0^1)(x*e^(x^2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the exponent x^2 is a multiple of the factor x outside the exponential.

2. Substitute u=x^2 which implies that the differential d(u)=2*x*d(x) or 1/2*d(u)=x*d(x)

3. Change the limits of integration by plugging the original x values into the substitution equation: when x=0 u=0^2=0 when x=1 u=1^2=1

4. Rewrite the integral in terms of u using the new limits and the differential substitution.

(∫_0^1)(1/2*e^u*d(u))

5. Integrate the exponential function, noting that the integral of e^u is e^u

1/2*[e^u]^1_0

6. Evaluate the expression at the upper and lower limits and subtract.

1/2*(e^1−e^0)

7. Simplify the result using the fact that e^0=1

1/2*(e−1)

Final Answer

(∫_0^1)(x*e^(x^2)*d(x))=(e−1)/2

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