Question

Which of the following is NOT a property of definite integral?
1. \(\displaystyle\int_a^b f(x) dx =- \displaystyle\int_b^a f(x) dx\)
2. \(\displaystyle\int_0^a f(x) dx = \displaystyle\int_0^a f(a-x) dx\)
3. \(\displaystyle\int_0^{2a} f(x) dx = \displaystyle\int_0^{2a} f(2a-x) dx\)
4. \(\displaystyle\int_a^b f(x) dx = - \displaystyle\int_a^b f(t) dt\)

Answer 1

Correct Answer - Option 4 : \(\displaystyle\int_a^b f(x) dx = - \displaystyle\int_a^b f(t) dt\)

Concept:

Property of definite integral: 

1) If f(x) is an odd function then,

 \(\int\limits_{ - a}^a {f(x)dx} = 0\)

2) If f(x) is an even function then,

 \(\int\limits_{ - a}^a {f(x)dx} =2\int\limits_0^a {f(x)dx}\)

3) \(\mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = \mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {{\rm{a}} - {\rm{x}}} \right){\rm{dx}}\)

 

4) \(\rm \int_a^bf(x)\ dx=\int_a^bf(a+b-x)\ dx\)

 

5) If f(x) = f(2a - x), then, \(\rm \int_0^{2a}f(x)\ dx=2\int_0^af(x)\ dx\)