The value of `int_(pi//2)^(pi//2)(x^3+x\ cos x+tan^5x+1)dx ,` is `2` b. `pi` c. `0` d. 1

Explanation

`int_{-frac{pi}{2}}^{frac{pi}{2}}(x^{3}+x cos x+tan ^{5} x+1) d x `

`=[frac{x^{4}}{4}]_{-frac{pi}{2}}^{frac{pi}{2}}+[x sin x]_{-frac{pi}{2}}^{frac{pi}{2}}-int_{-frac{pi}{2}}^{frac{pi}{2}} sin x d x+int_{-frac{pi}{2}}^{frac{pi}{2}} tan ^{3} x(sec ^{2} x-1) d x+[x]_{-frac{pi}{2}}^{frac{pi}{2}} `

`=frac{pi^{4}}{64}-frac{pi^{4}}{64}+frac{pi}{2}-frac{pi}{2}-[-cos x]_{-frac{pi}{2}}^{frac{pi}{2}}+int_{-frac{pi}{2}}^{frac{pi}{2}} tan ^{3} x sec ^{2} x d x-int_{-frac{x}{2}}^{frac{pi}{2}} tan ^{3} x d x+frac{pi}{2}+frac{pi}{2} `

`=pi+0+[frac{tan ^{4} x}{4}]_{-frac{pi}{2}}^{frac{pi}{2}}-int_{-frac{x}{2}}^{frac{pi}{2}} tan x sec ^{2} x d x-int_{-frac{pi}{2}}^{frac{x}{2}} tan x d x `

`=pi-[frac{tan ^{2} x}{2}]_{-frac{pi}{2}}^{frac{pi}{2}}-[-log (cos x)]_{-frac{pi}{2}}^{frac{pi}{2}} `

`=pi`