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cscott
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Homework Statement
Find the following intergral...
Homework Equations
[tex]\int{\frac{-1}{(x^2+1)^2}dx}[/tex]
The Attempt at a Solution
I can't get anywhere with this... any hints?
Last edited:
Solving integrals involving fractions can often be tricky, but in this case, we can use a substitution method. Let u = x^2 + 1, then du = 2x dx. Substituting this into the integral, we get -1/2 * ∫ du/u^2. This can be solved using the power rule for integrals, giving us the final answer of -1/2u + C. Remember to substitute back in for u to get the final answer in terms of x.
Unfortunately, there is no specific shortcut or trick to solving this integral. However, understanding and applying substitution, as shown in the previous answer, can often make solving integrals involving fractions easier.
The domain of this integral is all real numbers except for x = ±i, where i is the imaginary unit. This is because the original function has a singularity at these points, making the integral undefined.
No, integration by parts is not applicable in this case. Integration by parts is typically used when the integral contains a product of two functions, but in this case, we only have a single function -1/(x^2 +1)^2.
Yes, it is possible to graph the function y = -1/(x^2 +1)^2, which is the antiderivative of the integrand. However, keep in mind that the graph will have a discontinuity at x = ±i due to the singularity of the original function at these points.