求定积分 ∫(0~π/2)(cosx)^5*(sinx)^2 dx

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求定积分 ∫(0~π/2)(cosx)^5*(sinx)^2 dx

求定积分 ∫(0~π/2)(cosx)^5*(sinx)^2 dx
求定积分 ∫(0~π/2)(cosx)^5*(sinx)^2 dx

求定积分 ∫(0~π/2)(cosx)^5*(sinx)^2 dx
∫(0→π/2)(cosx)^5·(sinx)²dx
=∫(0→π/2)(cosx)^4·(sinx)²d(sinx)
=∫(0→π/2)(1-sin²x)²·(sinx)²d(sinx)
=∫(0→π/2)[(sinx)^6+2(sinx)^4+(sinx)^2]d(sinx)
=[1/3(sinx)^3-2/5(sinx)^5+1/7(sinx)^7] |(0→π/2)
=1/3-2/5+1/4
=8/105


∫(0→π/2)(cosx)^5·(sinx)²dx
=∫(0→π/2)(cosx)^4·(sinx)²d(sinx)
=∫(0→π/2)(1-sin²x)²·(sinx)²d(sinx)
=∫(0→π/2)[(sinx)^6+2(sinx)^4+(sinx)^2]d(sinx)
=[1/3(sinx)^3-2/5(sinx)^5+1/7(sinx)^7] |(0→π/2)
=1/3-2/5+1/4
=8/105