Integrate 1/(t^2+1)^2 Using i and Partial Fractions

In summary, the use of i in the integration of 1/(t^2+1)^2 allows for solving for the integral of complex-valued functions. To convert the fraction into partial fractions, the denominator must first be factored into linear factors. The general formula for integrating partial fractions involves solving for coefficients and then using ln|x-a| terms in the integration. The steps for integrating 1/(t^2+1)^2 using partial fractions include factoring, rewriting, solving for coefficients, and using the general formula. Special cases to consider include repeated linear factors and irreducible quadratic factors in the denominator.
  • #1
cragar
2,552
3
i was trying to integrate [itex] \frac{1}{(t^2+1)^2} [/itex]
By factoring it into [itex] \frac{1}{(t+i)^2(t-i)^2} [/itex] and then doing partial fractions.
then integrating each term using a u substitution. Ok but then how do I get the real part out of this solution. I know the arctan(t) can be extracted out of stuff of this form.
 
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  • #2
use
$$\arctan(x)=\tfrac{1}{2}\imath \, (\log(1-\imath\, x)-\log(1+\imath\, x))$$
 

FAQ: Integrate 1/(t^2+1)^2 Using i and Partial Fractions

1. What is the purpose of using i in the integration of 1/(t^2+1)^2?

The use of i, the imaginary unit, allows us to solve for the integral of complex-valued functions. In this case, it is used to represent the complex numbers in the denominator of the fraction.

2. How do you convert 1/(t^2+1)^2 into partial fractions?

To convert the fraction into partial fractions, we must first factor the denominator into linear factors. In this case, we have (t^2+1)^2 = (t^2+1)(t^2+1). Then, we can rewrite the fraction as A/(t+i) + B/(t-i) + C/(t^2+1) + D/(t^2+1)^2. From here, we can solve for the coefficients A, B, C, and D.

3. What is the general formula for the integration of partial fractions?

The general formula for the integration of partial fractions is ∫(A/(x-a) + B/(x-b) + ... + N/(x-n))dx = Aln|x-a| + Bln|x-b| + ... + Nln|x-n| + C, where A, B, ..., N are the coefficients and C is the constant of integration.

4. Can you explain the steps involved in integrating 1/(t^2+1)^2 using partial fractions?

First, we factor the denominator into linear factors: (t^2+1)^2 = (t+i)^2(t-i)^2. Then, we rewrite the fraction as A/(t+i) + B/(t-i) + C/(t^2+1) + D/(t^2+1)^2. Next, we solve for the coefficients A, B, C, and D. Finally, we use the general formula for the integration of partial fractions to integrate each of the terms, and add them together with the constant of integration to get the final solution.

5. Are there any special cases to consider when integrating 1/(t^2+1)^2 using partial fractions?

Yes, there are some special cases to consider. If the denominator has repeated linear factors, then we must include terms with higher powers of the linear factors in the partial fractions. In this case, we would have terms such as A/(t+i)^2 + B/(t-i)^2. Additionally, if the denominator has irreducible quadratic factors, then we must include terms with the linear factors and the irreducible quadratic factors. In this case, we would have terms such as A/(t+i) + B/(t-i) + C/(t^2+1) + D/(t^2+1)^2.

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