How do you integrate int x*2^x by integration by parts method?

1 Answer
Aug 3, 2016

=2^x/ (ln 2) ( x - 1/ (ln 2) ) + C

Explanation:

firstly know that d/dx ( a ^x ) = ln a \ a^x

So

int x*2^x \ dx

= int x d/dx( 1/ ln 2 2^x) \ dx

= x/ ln 2 2^x - int d/dx(x) 1/ ln 2 2^x\ dx

= x/ ln 2 2^x - 1/ ln 2 int 2^x\ dx

= x/ ln 2 2^x - 1/ ln 2 int d/dx( 1/ ln 2 2^x)\ dx

= x/ ln 2 2^x - 2^x 1/( ln 2 )^2+ C

=2^x/ (ln 2) ( x - 1/ (ln 2) ) + C