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

Problem

(∫_^)(x3*e(2*x)*d(x))

Solution

1. Identify the method of integration by parts, which states (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) We will use the tabular method (or repeated integration by parts) since we have a polynomial x3 multiplied by an exponential e(2*x)

2. Differentiate the polynomial part u=x3 until it reaches zero:

(u_0)=x3

(u_1)=3*x2

(u_2)=6*x

(u_3)=6

(u_4)=0

3. Integrate the exponential part d(v)=e(2*x)*d(x) the same number of times:

(v_0)=e(2*x)

(v_1)=1/2*e(2*x)

(v_2)=1/4*e(2*x)

(v_3)=1/8*e(2*x)

(v_4)=1/16*e(2*x)

4. Combine the terms by multiplying (u_n) by (v_n+1) and alternating signs starting with positive:

(x3)*(1/2*e(2*x))−(3*x2)*(1/4*e(2*x))+(6*x)*(1/8*e(2*x))−(6)*(1/16*e(2*x))

5. Simplify the coefficients and factor out the common term 1/2*e(2*x) or simply write the expanded form:

1/2*x3*e(2*x)−3/4*x2*e(2*x)+3/4*x*e(2*x)−3/8*e(2*x)+C

Final Answer

(∫_^)(x3*e(2*x)*d(x))=1/2*x3*e(2*x)−3/4*x2*e(2*x)+3/4*x*e(2*x)−3/8*e(2*x)+C

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