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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 5.4, Problem 32E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Evaluating a Definite Integral In Exercises 17-38, evaluate the definite integral. See Examples 3 and 4.

∫ 0 2 x 1 + 2 x 2 d x

To determine

To calculate: The value of definite integral ∫02x1+2x2dx.

Explanation

Given Information:

The integral is ∫02x1+2x2dx.

Formula used:

The fundamental theorem of calculus states that,

If f is integrable on interval [a,b] then ∫abf(x)dx=F(b)−F(a).

The integration formula is ∫xndx=xn+1n+1+C.

Calculation:

Consider the integral,

∫02x1+2x2dx

The integral in the radical form can be written as,

∫02x1+2x2dx=∫02x(1+2x2)−1/2dx

Multiply and divide by 4,

∫02x(1+2x2)−1/2dx=∫02x(1+2x2)−1/2(4)dx4

Now use the substitution procedure,

1+2x2=u4x=d(u)dx4xdx=du

The lower limit is,

1+2x2=u1+2(0)2=uu=1

The upper limit is,

1+2x2=u1+2(2)2=u

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