Find the integral?

Question

Given f(x)=x+1 for x<0, cos pi*x for x≥0, find ∫ f(x)dx from -1 to 1.

 

Answer: 1/2

Answer 1

Since the function f(x) is defined piecewise, the range of integration must be split into the intervals

[-1 , 0] and [0.1]

 

For the first interval, the integral, I1, is  I1 = ∫ (x+1) dx  from -1 to 0.   The anti-derivative is

(1/2) x² + x, so   I1 = 0 -  [ (1/2) (-1)² -1  ] =  1/2

 

For the second interval, the integral, I2, is  I2 = ∫ cos(π x) from 0 to 1.   The anti-derivative is (1/π)sin(x),

so I2 =  (1/π)sin(π) - (1/π) sin(0)  =  0

 

Thus the total integral I1+I2 =  1/2  + 0 = 1/2