Question

How to find integration??

Given that `int_0^4 h(x) dx = 5`, evaluate:

a) `int_0^4 2h(x) dx`

b) `int_0^4 h(x)-x dx`

c) `int_0^4 h(x) +1 dx`

Answer 1

a) 10
b) -3
c) 9

Explanation:

a) Remember that a constant can be written outside the integral. Therefore,:
`int_0^4 2h(x) dx=2*int_0^4 h(x) dx=2*5=10`

b) Remember that `int (f(x)+g(x))=int(f(x))+int(g(x))`

Therefore:
`int_0^4 h(x)-x dx=int_0^4 h(x)dx-int_0^4x dx`
`int_0^4x dx=[1/2x^2]_0^4=16/2-1/2*0=8`
`int_0^4 h(x)dx-int_0^4x dx=5-8=-3`

c) `int_0^4 h(x) +1 dx=int_0^4 h(x)dx +int_0^4 1 dx`
`int_0^4 1 dx=[x]_0^4=4-0=4`
`int_0^4 h(x)dx +int_0^4 1 dx=5+4=9`