how should I integrate 1/(x^2 +1)^5
According to my table of integrals, this one is a real mess.
I had to use a recursion relation. This is equivalent to using integration by parts several times. The first step yields
INT of 1/(x^2 +1)^5
= x/[8(x^2+1)^4]
+ (1/4) INT of 1/(x^2 +1)^4
In the next step you reduce the second integral to another function of x plus the integral of 1/(x^2 +1)^3 etc.
There will be a log or hyperbolic tangent term in the last step
Drwls, how did you derive the first step so that I can continue with the remainder of the steps?
To integrate the function 1/(x^2 + 1)^5, you can use integration techniques such as substitution and partial fractions. Here is a step-by-step guide on how to approach this integral:
Step 1: Recognize the form
Notice that the denominator (x^2 + 1)^5 involves a quadratic term and a constant term. To simplify the integral, we can use a substitution to express the integral in terms of a simpler variable.
Step 2: Substitution
Let's substitute u = x^2 + 1. By doing so, we can rewrite the integral in terms of u:
∫ 1/u^5 du.
Step 3: Simplify the integral
Now that we have the integral in terms of u, we can simplify it by applying the power rule for integration. The integral becomes:
∫ u^(-5) du.
Step 4: Integrate using the power rule
Using the power rule, we can integrate u^(-5) as follows:
∫ u^(-5) du = u^(-5+1) / (-5+1) + C
= -u^(-4)/4 + C.
Step 5: Substitute back in the original variable
Finally, replace u with its original expression (x^2 + 1) and simplify the integral:
-∫ (x^2 + 1)^(-4) / 4 dx + C.
That's it! You have successfully integrated 1/(x^2 + 1)^5 using the substitution method.