Question

let me know if anyone needs help in calculus!?

Answer 1

Integrate x^3 dx / (x^2+1)^3.?
Integrate in two ways.
a) by letting x=tan(theta)
b) by letting z= x^2+1
c)show that both results are the same.

Any help will be appreciated

Answer 2

x=tan D
x^2+1=(tan D)^2+1=(secD)^2,. then
Integral x^3 dx / (x^2+1)^3
Integral (tan D )^3 dx / (sec D)^3
Integral (sinD/cosD )^3 dx / (1/cos D)^3
Integral (sinD )^3 dx / (1)^3
Integral (sinD )^3 dx
Integral sinD(sinD )^2dx
Integral sinD[1-(cos D)^2dx]
Integral sinD([-(sinD(cosD)^2 dx]
let u=cosD, Du=-sinD dx
=Integral sinD-(u)^2du
=-cosD +u^3/3 +c
=cos D +((cosD)^3) /3 +C ans

Answer 3

Try this... What can you say about the sequence \[A _{n}=n(2^{1/2}-1)?\]. Suggestion: look at the \[\lim_{x \rightarrow \infty} (2^{1/n}-1)/(1/x)\].

Then decide if the series \[\sum_{n=1}^{\infty}A _{n}\] would converge or diverge? Why?