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

Problem

(∫_^)(x3*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 we have a polynomial x3 multiplied by a trigonometric function cos(x)

2. Differentiate the polynomial term 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 term cos(x) the same number of times:

d(v)=cos(x)

(v_1)=sin(x)

(v_2)=−cos(x)

(v_3)=−sin(x)

(v_4)=cos(x)

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

(+)*(x3)*(sin(x))

(−)*(3*x2)*(−cos(x))

(+)*(6*x)*(−sin(x))

(−)*(6)*(cos(x))

5. Simplify the resulting expression and add the constant of integration C

x3*sin(x)+3*x2*cos(x)−6*x*sin(x)−6*cos(x)+C

Final Answer

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

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