Evaluate the Integral

Problem

(∫_^)((x+1)*e(4*x2+8*x)*d(x))

Solution

1. Identify the substitution method as the appropriate technique because the derivative of the exponent 4*x2+8*x is a multiple of the factor (x+1)

2. Define the substitution variable u and calculate its differential d(u)

u=4*x2+8*x

d(u)/d(x)=8*x+8

d(u)=8*(x+1)*d(x)

3. Isolate the terms present in the integral to substitute them in terms of u

1/8*d(u)=(x+1)*d(x)

4. Substitute the expressions into the integral.

(∫_^)(eu⋅1/8*d(u))

5. Factor out the constant and integrate with respect to u

1/8*(∫_^)(eu*d(u))

1/8*eu+C

6. Back-substitute the original expression for u to find the final result.

1/8*e(4*x2+8*x)+C

Final Answer

(∫_^)((x+1)*e(4*x2+8*x)*d(x))=1/8*e(4*x2+8*x)+C

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