Question

The value ${\int }_{-2012}^{2012}(\sin \left(x^3\right)+x^5+1)dx$ is

• A. $2012$
• B. $2013$
• C. $0$
• D. $4024$

Answer 1

Answer: (D)

$4024$

Let $I={\int }_{-2012}^{2012}\left(\sin \right(x^3)+x^5+1)dx...\left(1\right)$

Let $t=-x$

$\therefore dt=-dx$

$\therefore I={\int }_{2012}^{-2012}-\left(\sin \right(-t^3)-t^5+1)dt$

$I={\int }_{2012}^{-2012}\left(\sin \right(t^3)+t^5-1)dt$

$\therefore I={\int }_{-2012}^{2012}-\left(\sin \right(x^3)+x^5-1)dx...\left(2\right)$

Adding equations $\left(1\right)$ and $\left(2\right)$, we have

$2I={\int }_{-2012}^{2012}2dx=2(2012-(-2012\left)\right)$

$\therefore I=4024$