设∫(tanx)^ndx,I(n)=1/(n-1)*(tanx)^(n-1)-I(n-2),并求∫(tanx)^5dx,上面的n是下标

Question

设∫(tanx)^ndx,I(n)=1/(n-1)*(tanx)^(n-1)-I(n-2),并求∫(tanx)^5dx,上面的n是下标

Answer 1

I(n) = tanⁿ⁻¹x/(n - 1) - I(n - 2)
∫ tan⁵x dx = I(5)
= tan⁵⁻¹x/(5 - 1) - I(3)
= (1/4)tan⁴x - ∫ tan³x dx
= (1/4)tan⁴x - [tan³⁻¹/(3 - 1) - I]
= (1/4)tan⁴x - (1/2)tan²x + ∫ tanx dx
= (1/4)tan⁴x - (1/2)tan²x + ∫ - d(cosx)/cosx
= (1/4)tan⁴x - (1/2)tan²x - ln|cosx| + C