Evaluate the Integral

Problem

(∫_^)((2+2*x)*e(4*x+2*x2)*d(x))

Solution

1. Identify the inner function of the composite term to use for substitution. Let u=4*x+2*x2

2. Differentiate u with respect to x to find d(u)

d(u)/d(x)=4+4*x

3. Rewrite the differential d(u) by factoring out a 2

d(u)=(4+4*x)*d(x)

d(u)=2*(2+2*x)*d(x)

4. Solve for the expression found in the integral, (2+2*x)*d(x)

1/2*d(u)=(2+2*x)*d(x)

5. Substitute u and 1/2*d(u) into the original integral.

(∫_^)(eu⋅1/2*d(u))

6. Integrate with respect to u

1/2*(∫_^)(eu*d(u))=1/2*eu+C

7. Back-substitute the original expression for u

1/2*e(4*x+2*x2)+C

Final Answer

(∫_^)((2+2*x)*e(4*x+2*x2)*d(x))=1/2*e(4*x+2*x2)+C

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