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
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+C∫sintdt=−cost+C intcostdt=sint+C∫costdt=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
=[-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