Single correct optionThe value of ∫−11x3cosx dx\displaystyle\int_{-1}^{1}x^3\cos x\,dx∫−11x3cosxdx isA. 000B. 2sin12\sin12sin1C. cos1\cos1cos1D. 111Show answer and solutionAnswer: (A) 000Let f(x)=x3cosxf(x)=x^3\cos xf(x)=x3cosx. Then f(−x)=−x3cosx=−f(x)f(-x)=-x^3\cos x=-f(x)f(−x)=−x3cosx=−f(x), so fff is odd and its integral over [−1,1][-1,1][−1,1] is 0, with no calculation needed. Trap: attempting by-parts and mis-evaluating, or treating the integrand as even and doubling ∫01\int_0^1∫01, which gives a non-zero value.SaveMy notesReport a mistakeMy notes on this pageSaved on this device. Sign in or create a free account to keep it on every device.← Previous Definite Integrals questionNext Definite Integrals question →