Question

The value of :

${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(x^3+x\cos x+{\tan }^{5}x+1)dx$

• A. $0$
• B. $2$
• C. $\pi$
• D. $1$

Answer 1

Answer: (C)

$\pi$

Let $I={\int }_{-\pi /2}^{\pi /2}(x^3+x\cos x+{\tan }^{5}x+1)dx$

We know,

If $f\left(x\right)$ is an even function, then ${\int }_{-a}^{a}f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx$

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

Thus, $I=0+0+0+2{\int }_0^{\pi /2}dx$

$I=2[x{]}_o^{\pi /2}=\frac{2\pi }{2}=\pi$