SOLUTION: Please help me solve this equation: Evaluate the following integrals. a) {{{ int ( lnt/sqrt(t), dt, 4, 9) }}} b) {{{ int ( cos(sqrt(6x)), dx ) }}}

Algebra ->  Test -> SOLUTION: Please help me solve this equation: Evaluate the following integrals. a) {{{ int ( lnt/sqrt(t), dt, 4, 9) }}} b) {{{ int ( cos(sqrt(6x)), dx ) }}}      Log On


   

Question 355599: Please help me solve this equation:
Evaluate the following integrals.
a) +int+%28+lnt%2Fsqrt%28t%29%2C+dt%2C+4%2C+9%29+
b) +int+%28+cos%28sqrt%286x%29%29%2C+dx+%29+
Found 2 solutions by nyc_function, robertb:
Answer by nyc_function(2741) About Me  (Show Source):
You can put this solution on YOUR website!
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Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
a) +int+%28+lnt%2Fsqrt%28t%29%2C+dt%2C+4%2C+9%29+
Use integration by parts.
Let u+=+lnt, and dv+=+dt%2Fsqrt%28t%29. Then
du+=+dt%2Ft, and v+=+2sqrt%28t%29

=2sqrt%28t%29%2Alnt+-+2int%28dt%2Fsqrt%28t%29%29
=2sqrt%28t%29%2Alnt+-+4sqrt%28t%29. Thus
=12ln3-8ln2-4.

b) +int+%28+cos%28sqrt%286x%29%29%2C+dx+%29+
Let w+=+sqrt%286x%29. Then dt+=+sqrt%286x%29dw%2F3.
The integral becomes %281%2F3%29%2Aint+%28wcosw%2C+dw%29. Then use integration by parts:
Let u = w, and dv = coswdw. Then
%281%2F3%29%2Aint+%28wcosw%2C+dw%29+=+%28wsinw-int%28sinw%2Cdw%29%29%2F3
=%28wsinw+%2B+cosw%29%2F3%2Bk
=%28sqrt%286x%29%2Asin%28sqrt%286x%29%29+%2B+cos%28sqrt%286x%29%29%29%2F3%2Bk