一道积分题目 170
积分函数：arcsin(x)/(x(x(1-x))^(1/2))上限1，下限1/2一楼的看清楚题目好不好...
积分函数：arcsin(x)/(x(x(1-x))^(1/2))
上限1，下限1/2
一楼的看清楚题目好不好 展开
上限1，下限1/2
一楼的看清楚题目好不好 展开
2个回答
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S f(x) d(g(x)) = f(x) g(x) - S g(x) d(f(x))
Also, d(arcsin(x)) = 1/(1-x^2)^(1/2) dx
So,
S arcsin(x)/(1-x)^(1/2) dx = 2 S arcsin(x) d((1-x)^(1/2)) = 2 arcsin(x) (1-x)^(1/2) - 2 S (1-x)^(1/2) d(arcsin(x)) = 2 arcsin(x) (1-x)^(1/2) - 2 S (1-x)^(1/2)/(1-x^2)^(1/2) dx = 2 arcsin(x) (1-x)^(1/2) - 2 S (1-x)^(1/2)/((1-x)^(1/2) (1+x)^(1/2)) dx = 2 arcsin(x) (1-x)^(1/2) - 2 S 1/(1+x)^(1/2) dx =
2 arcsin(x) (1-x)^(1/2) - 4(1+x)^(1/2)
Also, d(arcsin(x)) = 1/(1-x^2)^(1/2) dx
So,
S arcsin(x)/(1-x)^(1/2) dx = 2 S arcsin(x) d((1-x)^(1/2)) = 2 arcsin(x) (1-x)^(1/2) - 2 S (1-x)^(1/2) d(arcsin(x)) = 2 arcsin(x) (1-x)^(1/2) - 2 S (1-x)^(1/2)/(1-x^2)^(1/2) dx = 2 arcsin(x) (1-x)^(1/2) - 2 S (1-x)^(1/2)/((1-x)^(1/2) (1+x)^(1/2)) dx = 2 arcsin(x) (1-x)^(1/2) - 2 S 1/(1+x)^(1/2) dx =
2 arcsin(x) (1-x)^(1/2) - 4(1+x)^(1/2)
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S arcsin(x)/(1-x)^(1/2) dx = 2 S arcsin(x) d((1-x)^(1/2)) = 2 arcsin(x) (1-x)^(1/2) - 2 S (1-x)^(1/2) d(arcsin(x)) = 2 arcsin(x) (1-x)^(1/2) - 2 S (1-x)^(1/2)/(1-x^2)^(1/2) dx = 2 arcsin(x) (1-x)^(1/2) - 2 S (1-x)^(1/2)/((1-x)^(1/2) (1+x)^(1/2)) dx = 2 arcsin(x) (1-x)^(1/2) - 2 S 1/(1+x)^(1/2) dx =
2 arcsin(x) (1-x)^(1/2) - 4(1+x)^(1/2)
2 arcsin(x) (1-x)^(1/2) - 4(1+x)^(1/2)
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