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

Problem

(∫_^)(x3*sin(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 a trigonometric function sin(x)

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

(u_0)=x3

(u_1)=3*x2

(u_2)=6*x

(u_3)=6

(u_4)=0

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

(v_0)=sin(x)

(v_1)=−cos(x)

(v_2)=−sin(x)

(v_3)=cos(x)

(v_4)=sin(x)

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

Term 1=+(x3)*(−cos(x))=−x3*cos(x)

Term 2=−(3*x2)*(−sin(x))=3*x2*sin(x)

Term 3=+(6*x)*(cos(x))=6*x*cos(x)

Term 4=−(6)*(sin(x))=−6*sin(x)

5. Sum the terms and add the constant of integration C

Final Answer

(∫_^)(x3*sin(x)*d(x))=−x3*cos(x)+3*x2*sin(x)+6*x*cos(x)−6*sin(x)+C

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