高数题目第13题
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被积函数定义域为-4<x<4
令x=4sint(-π/2<t<π/2),则t=arcsin(x/4),dx=4costdt,√(16-x²)=4cost
原式=∫4cost*4costdt
=16∫cos²tdt
=8∫(1+cos2t)dt
=4∫(1+cos2t)d(2t)
=4∫d(2t)+4∫cos2td(2t)
=8t+4sin2t+C
=8arcsin(x/4)+8sintcost+C
=8arcsin(x/4)+2x*√(16-x²)/4+C
=8arcsin(x/4)+x/2*√(16-x²)+C
令x=4sint(-π/2<t<π/2),则t=arcsin(x/4),dx=4costdt,√(16-x²)=4cost
原式=∫4cost*4costdt
=16∫cos²tdt
=8∫(1+cos2t)dt
=4∫(1+cos2t)d(2t)
=4∫d(2t)+4∫cos2td(2t)
=8t+4sin2t+C
=8arcsin(x/4)+8sintcost+C
=8arcsin(x/4)+2x*√(16-x²)/4+C
=8arcsin(x/4)+x/2*√(16-x²)+C
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