How Do You Integrate (16x^4 - 4)/(4x^2+1) from 0 to 1?

[ddr]

Homework Statement

integral defined in 0 (down) and 1 (up) of:
(16x^4 - 4)/(4x^2+1)

Homework Equations

The Attempt at a Solution

maybe the partition mode?
who help me?

 

[neutrino]

Just wondering...by any chance, is the denominator of the integrand 4x²+2?

 

[arildno]

Hint:
Use polynomial division first.

 

[ddr]

i can use the substitution rule, with u=4x^2?

 

[HallsofIvy]

You could but it doesn't really help since du= 8xdx doesn't give you anything easy. The numerator obviously factors into (4x²-2)(4x²+2)- that's why arildno asked if the denominator wasn't actually 4x²+ 2 rather than 4x²+ 1. But the world is never that easy, not even homework problems.

Best thing to do is arildno's suggest. Go ahead and divide 16x⁴- 4 by 4x²+ 1. The result will be a cubic polynomial plus a linear term, Ax+ B, over 4x²+ 1. To integrate Ax/(4x²+1), let u= 4x²+ 1. To integrate B/(4x²+1), use the arctangent.