Evaluate the Integral integral of xe^(4x) with respect to x

Problem

(∫_^)(x*e(4*x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function x and an exponential function e(4*x) use integration by parts.

2. Choose the parts for the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x and d(v)=e(4*x)*d(x)

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

d(u)=d(x)

v=(∫_^)(e(4*x)*d(x))=1/4*e(4*x)

4. Apply the integration by parts formula.

(∫_^)(x*e(4*x)*d(x))=x*(1/4*e(4*x))−(∫_^)(1/4*e(4*x)*d(x))

5. Evaluate the remaining integral.

(∫_^)(1/4*e(4*x)*d(x))=1/4⋅1/4*e(4*x)=1/16*e(4*x)

6. Combine the terms and add the constant of integration C

1/4*x*e(4*x)−1/16*e(4*x)+C

Final Answer

(∫_^)(x*e(4*x)*d(x))=1/4*x*e(4*x)−1/16*e(4*x)+C

Want more problems? Check here!