How do you evaluate the definite integral int (sint+cost)dt(sint+cost)dt from [0,pi/4][0,π4]?

1 Answer
Nov 6, 2016

int_0^(pi/4)(sint+cost)dt=1π40(sint+cost)dt=1

Explanation:

We have to know that:

  • int(f(x)+g(x))dx=intf(x)dx+intg(x)dx(f(x)+g(x))dx=f(x)dx+g(x)dx
  • intsintdt=-cost+Csintdt=cost+C
  • intcostdt=sint+Ccostdt=sint+C

Combining these and keeping the bounds of the integral, we see that:

int_0^(pi/4)(sint+cost)dt=[-cost+sint]_0^(pi/4)π40(sint+cost)dt=[cost+sint]π40

Evaluating at t=pi/4t=π4 then subtracting the function evaluated at t=0t=0:

=[-cos(pi/4)+sin(pi/4)]-[-cos0+sin0]=[cos(π4)+sin(π4)][cos0+sin0]

=(-sqrt2/2+sqrt2/2)-(-1+0)=(22+22)(1+0)

=0+1=0+1

=1=1