Evaluate the Integral

Problem

(∫_^)((1−6*x)*e(3*x−9*x2)*d(x))

Solution

1. Identify a potential substitution by looking for a function whose derivative appears in the integrand.

2. Let u be the exponent of the natural exponential function.

u=3*x−9*x2

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

d(u)/d(x)=3−18*x

d(u)=(3−18*x)*d(x)

4. Factor the expression for d(u) to match the terms present in the integral.

d(u)=3*(1−6*x)*d(x)

1/3*d(u)=(1−6*x)*d(x)

5. Substitute u and 1/3*d(u) into the original integral.

(∫_^)(eu⋅1/3*d(u))

6. Evaluate the integral using the rule (∫_^)(eu*d(u))=eu+C

1/3*(∫_^)(eu*d(u))=1/3*eu+C

7. Substitute back the original expression for u to get the final result in terms of x

1/3*e(3*x−9*x2)+C

Final Answer

(∫_^)((1−6*x)*e(3*x−9*x2)*d(x))=(e(3*x−9*x2))/3+C

Want more problems? Check here!