Chapter 8.3, Problem 14E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Finding an Indefinite Integral Involving Sine and Cosine In Exercises 3–14, find the indefinite integral. ∫ x 2 sin 2 x d x

To determine

To calculate: the value of the indefinite integral x2sin2xdx

Explanation

Given: The indefinite integral âˆ«x2sin2xdx

Formula Used:

cos2t+sin2t=1.

cos2x=1âˆ’2sin2x.

âˆ«cos2xdx=12sin2x+C.

âˆ«(sin2x)dx=âˆ’12cos2x+C.

ddxxn=nxnâˆ’1.

âˆ«xndx=xn+1n+1+c

Calculation:

Integrating by parts and letting u=x2 ----------(1), and

dv=sin2xdx ---------(2)

Applying the trigonometric identity cos2t=1âˆ’2sin2t, equation (2) becomes

dv=(1âˆ’cos2x2)dx ------------(3)

Integrating equation (3), we get

âˆ«dv=âˆ«(1âˆ’cos2x2)dxâˆ«dv=12âˆ«dxâˆ’12âˆ«(cos2x)dx

âˆ«dv=x2âˆ’14sin2x+C ---------------(4)

Now,

âˆ«x2sin2xdx=x2âˆ«sin2xdxâˆ’âˆ«(ddxx2âˆ«sin2xdx)dx.

âˆ«x2sin2xdx=x2âˆ«dvâˆ’âˆ«(2xâˆ«dv)dx -------(5)

Putting the result of (4) in equation (5), we get

âˆ«x2sin2xdx=x2(x2âˆ’14sin2x+C)âˆ’âˆ«(2x(x2âˆ’14sin2x+C))dx.

âˆ«x2sin2xdx=x32âˆ’x24sin2xâˆ’âˆ«(x2âˆ’x2sin2x)dx

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