Question

Evaluate: ${\int }_{-2}^2(x^{11}\cos x+e^x)dx$

• A. $\sinh 2$
• B. $2\sinh 2$
• C. $\frac{3}{2}\sinh 2$
• D. $\frac{\sinh 2}{2}$

Answer 1

The correct option is B $2\sinh 2$

${\int }_{-2}^2(x^{11}\cos x+e^x)dx$

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

And, $x^{11}\cos x$ is an odd function,

hence ${\int }_{-2}^2\left(x^{11}\cos x\right)dx=0$

Now, ${\int }_{-2}^2{e}^{x}dx$

$={\left[e^x\right]}_{-2}^2$

$=(e^2-e^{-2})$

$=2\times \frac{e^2-e^{-2}}{2}$

And, $\sinh z=\frac{e^z-e^{-z}}{2}$

Hence, ${\int }_{-2}^2(x^{11}\cos x+e^x)$

$=2\times \frac{e^2-e^{-2}}{2}$

$=2\sinh 2$