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

  • Thread starter ddr
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    Integral
In summary, the problem involves finding the integral of (16x^4 - 4)/(4x^2+1) from 0 to 1. The suggested approach is to use polynomial division first and then use substitution and the arctangent function to integrate.
  • #1
ddr
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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?
 
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  • #2
Just wondering...by any chance, is the denominator of the integrand 4x2+2?
 
Last edited:
  • #3
Hint:
Use polynomial division first.
 
  • #4
i can use the substitution rule, with u=4x^2?
 
  • #5
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.
 

1. What is an integral?

An integral is a mathematical concept that represents the accumulation of a quantity over a given interval. It is often described as the inverse operation of differentiation and can be used to find the area under a curve or to solve various physical problems.

2. How do I solve an integral?

To solve an integral, you need to first identify the function to be integrated and the limits of integration. Then, you can use different techniques such as substitution, integration by parts, or trigonometric substitution to find the antiderivative of the function. Finally, you can evaluate the antiderivative at the limits of integration to find the definite integral.

3. What are the different types of integrals?

There are two main types of integrals: definite and indefinite. A definite integral has specific limits of integration and gives a numerical value. An indefinite integral has no limits of integration and represents a family of functions with a constant of integration.

4. What is the purpose of solving an integral?

Solving an integral has various applications in mathematics, physics, and engineering. It can be used to find the area under a curve, calculate volumes and surfaces of revolution, find the average value of a function, and solve differential equations.

5. What are some tips for solving integrals?

Some tips for solving integrals include: identifying the type of integral, using appropriate substitution or integration techniques, checking for symmetry or special properties, and practicing with a variety of examples. It is also helpful to have a good understanding of algebra and trigonometry to simplify the integrand.

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