Question

高数中的一个三角函数定积分要怎么求？有什么简便方法吗？

Answer 1

追答

结果2/5

Answer 2

原式=2∫(0,π/2)sin^5θdθ
=2×4/5 ×2/3×1
=16/15.

追问

这是怎么变换的？

Answer 3

∫(0,π)sin^5θdθ
=-∫(0,π) sin^4θdcosθ
=-∫(0,π) (1-cos^2θ)^2dcosθ
=-∫(0,π) (1-2cos^2θ+cos^4θ)dcosθ
=-[cosθ-2/3cos^3θ+1/5cos^5θ]|(0,π)
=-[(cosπ-cos0)-2/3(cos^3 π-cos^3 0)+1/5(cos^5 π-cos^5 0)]
=-(-1-1)+2/3[(-1)^3-1^3]-1/5[(-1)^5-1^5]
=2-4/3+2/5
=30/15-20/15+6/15
=16/15

Answer 4

∫<0,π>(sinθ)^5dθ=∫<0,π>-(sinθ)^4dcosθ
=-∫<0,π>((1-(cosθ)^2)^2)dcosθ
=-∫<0,π>((1-2(cosθ)^2+(cosθ)^4)dcosθ
=-(cosθ-2/3(cosθ)^3+1/5(cosθ)^5)|θ=<0,π>
=-(-1-2/3*(-1)^3+1/5*(-1)^5)+(1-2/3*1^3+1/5*1^5)
=-(-1+2/3-1/5)+1-2/3+1/5
=16/15