Find-the-integral-xcosh-lnx-n-dx-where-n-0-1-2-

Question Number 2069 by Yozzi last updated on 01/Nov/15

F i n d t h e i n t e g r a l

\int {(x c o s h (l n x))}^{n} d x

w h e r e n = 0, 1, 2, \dots

Commented by prakash jain last updated on 01/Nov/15

\cosh (\ln x) = \frac{1}{2} (x + \frac{1}{x})

x \cdot \cosh (\ln x) = \frac{x^{2} + 1}{2}

I = \int {(\frac{x^{2} + 1}{2})}^{n} d x

Answered by prakash jain last updated on 01/Nov/15

I = \frac{1}{2^{n}} \int {(1 + x^{2})}^{n} d x

= \frac{1}{2^{n}} \int (^{n} C_{0} +^{n} C_{1} x^{2} +^{n} C_{2} x^{4} + \dots . +^{n} C_{n} x^{2 n}) d x

= \frac{1}{2^{n}} (^{n} C_{0} x +^{n} C_{1} \frac{x^{3}}{3} + \dots . +^{n} C_{n} \frac{x^{2 n + 1}}{2 n + 1})