notgoodatmath777
New member
- Joined
- Nov 1, 2020
- Messages
- 4
Sorry this is long, I'm mostly confused on parts b and c but would appreciate checking correctness on all parts.
a) I=0.
I did a leading order term analysis and saw that the integrand behave like 1/x^2 which converges by a p-test with p=2>1 and converges to 0 because as x approaches infinity, the denominator gets larger and larger making the numerator very small.
b) Integration by parts form is uv - integral of u'v.
u=(1+x^2)^n u' = 2nx(1+x^2)^(n-1) v'=1. v=x
I solved it out but I cannot seem to figure out where the -1/2n came from which is the uv part. I got uv = -x(1+x^2)^n
c) x^2 = A + B(1+x^2) I multiplied both sides by (1+x^2)^(n+1)
But I cannot find values of x for which A or B is equal to 0 to solve the other.
d) I=0 I feel like it's wrong
I plugged n=3 into the original equation to get the integral of 1/(1+x^2)^3. = 1/(1+x^6). and I did the same thing as part 1 to get that it converges to 0 by the p-test with p=6>1 because denominator gets very large so the numerator gets smaller.
a) I=0.
I did a leading order term analysis and saw that the integrand behave like 1/x^2 which converges by a p-test with p=2>1 and converges to 0 because as x approaches infinity, the denominator gets larger and larger making the numerator very small.
b) Integration by parts form is uv - integral of u'v.
u=(1+x^2)^n u' = 2nx(1+x^2)^(n-1) v'=1. v=x
I solved it out but I cannot seem to figure out where the -1/2n came from which is the uv part. I got uv = -x(1+x^2)^n
c) x^2 = A + B(1+x^2) I multiplied both sides by (1+x^2)^(n+1)
But I cannot find values of x for which A or B is equal to 0 to solve the other.
d) I=0 I feel like it's wrong
I plugged n=3 into the original equation to get the integral of 1/(1+x^2)^3. = 1/(1+x^6). and I did the same thing as part 1 to get that it converges to 0 by the p-test with p=6>1 because denominator gets very large so the numerator gets smaller.