Evaluate the Integral

Problem

(∫_^)((x2−5*x)*ex*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 the integrand is a product of a polynomial and an exponential function.

2. Assign the polynomial part to u and the exponential part to d(v) Let u=x2−5*x and d(v)=ex*d(x)

3. Differentiate u repeatedly until it reaches zero:

(u_0)=x2−5*x

(u_1)=2*x−5

(u_2)=2

(u_3)=0

4. Integrate d(v) repeatedly:

(v_0)=ex

(v_1)=ex

(v_2)=ex

(v_3)=ex

5. Combine the terms using alternating signs (+,−,+)

(∫_^)((x2−5*x)*ex*d(x))=(x2−5*x)*ex−(2*x−5)*ex+(2)*ex+C

6. Factor out the common term ex

(∫_^)((x2−5*x)*ex*d(x))=ex*(x2−5*x−2*x+5+2)+C

7. Simplify the expression inside the parentheses:

(∫_^)((x2−5*x)*ex*d(x))=ex*(x2−7*x+7)+C

Final Answer

(∫_^)((x2−5*x)*ex*d(x))=(x2−7*x+7)*ex+C

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