作业帮求不定积分∫x^2·√(4-x^2)dx
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求不定积分∫x^2·√(4-x^2)dx
求不定积分∫x^2·√(4-x^2)dx
求不定积分∫x^2·√(4-x^2)dx
令x=2sint,dx=2costdt
代入原式得：=∫4sint^2.2cost.2costdt
=16∫sint^2.cost^2dt
=4∫sin2t^2dt
=4∫[1-(cos4t-1)/2]dt
=6t-sin4t/2+c
t=arcsinx/2代入,得
2arcsin(x/2)-x/2*√(4-x^2)+C
第一方法：
∫x²/√(4-x²)dx (三角换元，令x=2sint)
=∫4(sint)^2/√(4(cost)^2)d(2sint)
=∫4(sint)^2/(2cost)*(2cost)dt
=∫4(sint)^2dt (倍角公式 cos2t=1-2(sint)^2)
=∫2(1-cos2t)dt
=2t-sin2...
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第一方法：
∫x²/√(4-x²)dx (三角换元，令x=2sint)
=∫4(sint)^2/√(4(cost)^2)d(2sint)
=∫4(sint)^2/(2cost)*(2cost)dt
=∫4(sint)^2dt (倍角公式 cos2t=1-2(sint)^2)
=∫2(1-cos2t)dt
=2t-sin2t+C (将 t=arcsin(x/2)带回)
=2arcsin(x/2)-2(x/2)*√(1-x^2/4)+C
=2arcsin(x/2)-x/2*√(4-x^2)+C
C为任意常数。
第二方法
>> sym x;
>> simple(int(x^2/sqrt(4-x^2),x))
simplify:
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
radsimp:
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
combine(trig):
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
factor:
-1/2*x*(-(x-2)*(x+2))^(1/2)+2*asin(1/2*x)
expand:
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
combine:
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
convert(exp):
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
convert(sincos):
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
convert(tan):
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
collect(x):
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
mwcos2sin:
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
ans =
-1/2*x*(4-x^2)^(1/2)+2*asin(1/2*x)
ans即为答案.
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