1 / | | 2*x /x\ | e *cos|-| dx | \2/ | / 0
Integral(exp(2*x)*cos(x/2), (x, 0, 1))
Use integration by parts, noting that the integrand eventually repeats itself.
For the integrand :
Let and let .
Then .
For the integrand :
Let and let .
Then .
Notice that the integrand has repeated itself, so move it to one side:
Therefore,
Now simplify:
Add the constant of integration:
The answer is:
/ 2*x /x\ /x\ 2*x | 2*e *sin|-| 8*cos|-|*e | 2*x /x\ \2/ \2/ | e *cos|-| dx = C + ------------- + ------------- | \2/ 17 17 | /
2 2 8 2*e *sin(1/2) 8*cos(1/2)*e - -- + ------------- + ------------- 17 17 17
=
2 2 8 2*e *sin(1/2) 8*cos(1/2)*e - -- + ------------- + ------------- 17 17 17
-8/17 + 2*exp(2)*sin(1/2)/17 + 8*cos(1/2)*exp(2)/17
Use the examples entering the upper and lower limits of integration.