Alexx1 said:[tex]\int \frac{dx}{(1+x^2)^n} \;=\; \frac{1}{2(n-1)}\cdot\frac{x}{(1+x^2)^{n-1}}
\;\;-\;\; \frac{2n-3}{2(n-1)}\int \frac{dx}{(1+x^2)^{n-1}}[/tex]
Can someone prove this?
berkeman said:What is the context of your question? Is it for schoolwork/homework?
Alexx1 said:I have exam (university) January 15th and we have exercises but we don't have answers.. so I would lik to know how to solve this one
Alexx1 said:[tex]\int \frac{dx}{(1+x^2)^n} \;=\; \frac{1}{2(n-1)}\cdot\frac{x}{(1+x^2)^{n-1}}
\;\;-\;\; \frac{2n-3}{2(n-1)}\int \frac{dx}{(1+x^2)^{n-1}}[/tex]
Can someone prove this?
The basic technique for proving the integral of (1+x^2)^n is to use the binomial theorem to expand the expression into a sum of terms, and then apply the power rule for integration to each term.
Yes, the integral of (1+x^2)^n can be evaluated using substitution. In particular, the substitution u = 1+x^2 can be used to transform the integral into a form that can be integrated using the power rule.
Yes, there are two special cases to consider when proving the integral of (1+x^2)^n. The first is when n = -1, in which case the integral will involve a natural logarithm. The second is when n is a negative integer, in which case the integral will involve a rational function.
The integral of (1+x^2)^n can be used to calculate the area under a curve by setting up the integral as the difference between the antiderivative evaluated at the upper and lower limits of integration. This will give the area between the curve and the x-axis.
Yes, the integral of (1+x^2)^n can be evaluated using integration by parts, but it may not lead to a simpler expression. In most cases, using the binomial theorem and the power rule will be a more efficient method for evaluating the integral.