Evaluate the Integral integral of 2xe^(3x) with respect to x

Problem

(∫_^)(2*x*e(3*x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function 2*x and an exponential function e(3*x) use integration by parts.

2. Assign variables for the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=2*x and d(v)=e(3*x)*d(x)

3. Differentiate u and integrate d(v) to find d(u) and v

d(u)/d(x)=2⇒d(u)=2*d(x)

v=(∫_^)(e(3*x)*d(x))=1/3*e(3*x)

4. Apply the integration by parts formula.

(∫_^)(2*x*e(3*x)*d(x))=(2*x)*(1/3*e(3*x))−(∫_^)((1/3*e(3*x))*(2)*d(x))

5. Simplify the expression and the remaining integral.

(∫_^)(2*x*e(3*x)*d(x))=2/3*x*e(3*x)−2/3*(∫_^)(e(3*x)*d(x))

6. Evaluate the final integral.

(∫_^)(e(3*x)*d(x))=1/3*e(3*x)

7. Combine all terms and add the constant of integration C

(∫_^)(2*x*e(3*x)*d(x))=2/3*x*e(3*x)−2/3*(1/3*e(3*x))+C

(∫_^)(2*x*e(3*x)*d(x))=2/3*x*e(3*x)−2/9*e(3*x)+C

Final Answer

(∫_^)(2*x*e(3*x)*d(x))=2/3*x*e(3*x)−2/9*e(3*x)+C

Want more problems? Check here!