   Chapter 8.4, Problem 33E

Chapter
Section
Textbook Problem

# Finding an Indefinite Integral In Exercises 19-32, find the indefinite integral. ∫ 1 4 + 4 x 2 + x 4   d x

To determine

To calculate: The solution of the indefinite integral 14+4x2+x4dx.

Explanation

Given:

The indefinite integral is 14+4x2+x4dx=1(x2+2)2dx.

Formula used:

The trigonometric identity cos2θ=1+cos2θ2.

The integral identity cosθdθ=sinθ+C.

Calculation:

Because x2+22 is of the form a2+x2. So, make use of the trigonometric solution, x=2tanθ to solve it as shown in figure.

Differentiate the above equation is;

dx=2sec2θdθ

Therefore;

1(x2+2)2dx=1((2tanθ)2+2)22sec2θdθ=1(2sec2θ)22sec2θdθ=122sec2θdθ=cos2θ22dθ

Further steps show that;

cos2θ22dθ=122(1+cos2θ)2dθ=142(1+cos2θ)dθ=142(θ+sin2θ2)+C=142(θ+2sinθcosθ2)+C

Further steps show that;

142

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