∫√x4+x6 dx

Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter6: Fractions
Section6.1: Simplifying Fractions
Problem 31WE
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∫√x4+x6 dx
Expert Solution
Step 1

Given that :

The integral is x4 +x6 dx .

To find:

Value of the integral.

Step 2

We can write, 

x4 +x6 dx = x4(1+x2 ) dx

                         = x2x2+1 dx

Substitute x = tan (u), then dx = sec2 (u) du.

Then,

x2x2+1 dx = sec3(u) . tan2(u) du

                          = sec3(u) . (-1+ sec2(u) )du=(-sec3(u)+ sec5(u) )du = -sec3(u) du + sec5(u) du 

Step 3

sec3u du = sec2(u) . sin(u)2 + 12  sec(u) du                       =sec2(u) . sin(u)2 + 12 ln tan (u) + sec(u)                       = 12  sec(u) .tan(u) + 12 ln tan (u) + sec(u)

sec5 (u) du = sec4(u) sin(u)4 + 34  sec3(u)du                          = sec4(u) sin(u)4 + 34 12  sec(u) .tan(u) + 12 ln tan (u) + sec(u)

Substitute back x = tan (u) ,then u = arctan(x),

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