   Chapter 4.5, Problem 48E

Chapter
Section
Textbook Problem

# Evaluate the definite integral. ∫ 0 4 x 1 + 2 x d x

To determine

To evaluate:

The given definite integral 04x1+2x dx

Explanation

1) Concept:

i) The substitution rule for definite integral:

If g'(x) is a continuous function on a,b whose f is continuous on range of u=g(x), then abfgxg'(x)dx=g(a)g(b)f(u)du. Here,g(x) is substituted as u and then the differentiation of g(x)dx =du.

ii) Indefinite integral

xn dx=xn+1n+1+C   (n-1)

iii)

ab[fx+gx] dx=abfxdx+abgxdx

iv)

abcfxdx=cabfxdx

2) Given:

04x1+2x dx

3) Calculation:

The given integral is

04x1+2x dx

Use the substitution method.

Substitute 1+2x=u   so  x=u-12. Differentiating with respect to x

dx=12du

The limits change and the new limits of integration are calculated by substituting

for x=0, u=1+2(0)=1 & for x=4, u=1+2(4) =9

Therefore, the given integral becomes

04x1+2x dx=19u-12udu2

Factor out 14 from the integration

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