SOLUTION: Please help me solve this equation. Evaluate the following definite integral: {{{ int( 2(cos y+1/y), dy, 1, pi ) }}}

Algebra ->  Test -> SOLUTION: Please help me solve this equation. Evaluate the following definite integral: {{{ int( 2(cos y+1/y), dy, 1, pi ) }}}      Log On


   

Question 344493: Please help me solve this equation. Evaluate the following definite integral: +int%28+2%28cos+y%2B1%2Fy%29%2C+dy%2C+1%2C+pi+%29+
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
+int%28+2%28cos+y%2B1%2Fy%29%2C+dy%2C+1%2C+pi+%29+
Using some basic properties of integrals we can factor out the 2:
+2%2Aint%28+%28cos+y%2B1%2Fy%29%2C+dy%2C+1%2C+pi+%29+
and separate it into two:
+2%2A%28int%28+cos%28+y%29%2C+dy%2C+1%2C+pi%29+%2B+int%28+1%2Fy%2C+dy%2C+1%2C+pi%29+%29+
Now we have two fairly simple integrals. Since cos(y)dy integrates to sin(y) and (1/y)dy integrates to ln(y) we should get:
+2%2A%28%28sin%28pi%29+-+sin%281%29%29+%2B+%28ln%28pi%29+-+ln%281%29%29%29
Since sin%28pi%29+=+0 and ln(1) = 0:
+2%2A%28%280+-+sin%281%29%29+%2B+%28ln%28pi%29+-+0%29%29
Simplifying:
+2%2A%28%28-+sin%281%29%29+%2B+%28ln%28pi%29%29%29
+-2sin%281%29+%2B+2ln%28pi%29

If you need a decimal answer, then make sure your calculator is set to radian mode when you find sin(1).