NIMCET 2020 — Mathematics PYQ

Concept:

For an odd function $f(x)$: ${\int }_{-a}^{a}f(x)dx=0$. 

Calculation:

We observe that $a{x}^5$ and $b{x}^3$ are odd functions of $x$, because $a(-x{)}^5=-a{x}^5$ and $b(-x{)}^3=-b{x}^3$.

$\therefore {\int }_{-2}^{2}a{x}^{5}dx=0$ and ${\int }_{-2}^{2}b{x}^{3}dx=0$.

The value of ${\int }_{-2}^2(a{x}^5+b{x}^3+c)dx$ depends on the value of c.

In fact, ${\int }_{-2}^2(a{x}^5+b{x}^3+c)dx={\int }_{-2}^{2}cdx=c[x{]}_{-2}^2=c[2-(-2)]=4c$.