Correct Answer - Option 4 : 0
Concept:
The function f(x) is an odd function if f(x) = - f(-x) and an even function if f(x) = f(-x).
- When f(x) is an even function then \(\rm \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\)
- When f(x) is odd function then \(\rm \int_{-a}^{a}f(x)dx=0\)
Calculation:
Given: \(\rm \int_{-5 }^5 x^{3} dx\)
Let f(x) = x3
As we can see that, f(- x) = (-x)3 = - x3 = - f(x).
So, f(x) = x3 is an odd function.
As we know that, when f(x) is an odd function then \(\rm \int_{-a}^{a}f(x)dx=0\)
\(\Rightarrow\rm \int_{-5 }^5 x^{3} dx=0\)
Hence, the correct option is 4.