Question

请大佬们解答下高数题？

怎么得到的

Answer 1

ln(x+√(1+x^2))为奇函数，
同时，积分区间为对称区间，
所以其积分为0.
则原积分=∫(-π/4,π/4)(sinx)^8dx
∫(sinx)^8dx
=-∫ (sinx)^7 dcosx
= -cosx (sinx)^7 + ∫ 7(cosx)^2(sinx)^6 dx
=-cosx (sinx)^7 +7∫ (1- (sinx)^2)(sinx)^6 dx
8∫(sinx)^8dx= -cosx (sinx)^7 + 7∫(sinx)^6dx
∫(sinx)^8dx
= (1/8) [-cosx (sinx)^7 + 7∫(sinx)^6dx]
= (1/8) {-cosx (sinx)^7 + (7/6)[-cosx (sinx)^5 + 5∫(sinx)^4dx]}
=(1/8){ -cosx (sinx)^7 + (7/6)(-cosx (sinx)^5 + (5/4)[-cosx(sinx)^3+3∫(sinx)^2dx]) }
=(1/8){ -cosx (sinx)^7 + (7/6)(-cosx (sinx)^5 + (5/4)[-cosx(sinx)^3+(3/2)[x-sin2x/2]) }
带入积分限即可…

Answer 2

因后项是奇函数，在对称区间积分为 0.