Chapter 8.2, Problem 63E

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

Chapter
Section

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

# Using Two Methods Integrate ∫ x 3 4 + x 2 d x (a) by parts, letting d v = x 4 + x 2 d x (b) by substitution, letting u = 4 + x 2 .

(a)

To determine

To calculate: The expression for the indefinite integral x34+x2dx by integration by parts.

Explanation

Given:

The granted indefinite integral is x34+x2dx and take dv=x4+x2dx

Formula used:

The integration of:

tndt=tn+1n+1

The integration by parts:

udv=uvvdu

The differentiation of ddxxn=nxn1

Calculation:

To integrate x34+x2dx by parts, choose u=x2, so that dv=x4+x2dx

Now, integrate on both sides as follows:

dv=x4+x2dx

dv=x4+x2dx …...…... (1)

Put 4+x2=t and differentiate with respect to x as follows:

ddx(4+x2)=dtdx2x=dtdx2xdx=dt

(x)dx=12dt …...…... (2)

Substitute (2) in (1) as follows:

dv=12dttdv=12t12dtdv=12t12+1(12+1)+Cdv=t+C

dv=4+x2+C …...…... (3)

Further, evaluating as follows:

x34+x2dx=(x2)(x4+x2dx)=(x2)(dv)

(b)

To determine

To calculate: The expression for the indefinite integral x34+x2dx by substitution method.

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