Evaluate the Integral

Problem

(∫_^)((x2+8*x)*cos(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 the algebraic part (x2+8*x) will eventually differentiate to zero.

2. Differentiate the algebraic part u=x2+8*x repeatedly until it reaches zero.

(u_0)=x2+8*x

(u_1)=2*x+8

(u_2)=2

(u_3)=0

3. Integrate the trigonometric part d(v)=cos(x)*d(x) repeatedly.

(v_0)=cos(x)

(v_1)=sin(x)

(v_2)=−cos(x)

(v_3)=−sin(x)

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

Term 1=+(x2+8*x)*(sin(x))

Term 2=−(2*x+8)*(−cos(x))=+(2*x+8)*cos(x)

Term 3=+(2)*(−sin(x))=−2*sin(x)

5. Simplify the expression by combining the sine terms and adding the constant of integration C

(x2+8*x−2)*sin(x)+(2*x+8)*cos(x)+C

Final Answer

(∫_^)((x2+8*x)*cos(x)*d(x))=(x2+8*x−2)*sin(x)+(2*x+8)*cos(x)+C

Want more problems? Check here!