Evaluate the Integral

Problem

(∫_0^2)(8*x*e(x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the exponent x2 has a derivative 2*x that is a factor of the integrand.

2. Substitute u=x2 which implies that the differential d(u)=2*x*d(x)

3. Adjust the integrand by rewriting 8*x as 4⋅2*x to match the differential d(u)

4. Change the limits of integration: when x=0 u=0=0 when x=2 u=2=4

5. Rewrite the integral in terms of u

(∫_0^4)(4*eu*d(u))

6. Integrate the expression with respect to u

4*[eu]40

7. Evaluate the definite integral by plugging in the upper and lower limits:

4*(e4−e0)

8. Simplify the result using the fact that e0=1

4*(e4−1)

Final Answer

(∫_0^2)(8*x*e(x2)*d(x))=4*e4−4

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