新疆
求I=∫95x^3/√(42-6x^2)dx的不定積分
主要內容：
本文根據分部積分法、三角換元法以及湊分法等方法，介紹不定積分I=∫95x^3/√(42-6x^2)dx的計算步驟。
解法一：
思路：根據分子分母的關系，直接變形化簡使用湊分法求得：
I=-∫(95/6)[x(42-6x^2)-42x]dx/√(42-6x^2)
=-(95/6)∫x(42-6x^2)dx/√(42-6x^2)+ 665∫xdx/√(42-6x^2)
=-(95/6)∫x√(42-6x^2)dx-1995/6^2∫d(42-6x^2)/√(42-6x^2)
=-95/6^2∫√(42-6x^2)d(42-6x^2)-3990/6^2√(42-6x^2)
=(190/3) /6^2√(42-6x^2)^3-3990/6^2*√(42-6x^2)+c
解法二：
思路：利用不定積分的分部積分方法求得：
I=95∫x^2*xdx/√(42-6x^2)
=-(95/12)∫x^2d(42-6x^2)/√(42-6x^2)
=-(95/12)∫x^2d√(42-6x^2)
=-(95/12)x^2√(42-6x^2)+(95/12) ∫√(42-6x^2)dx^2
=-(95/12)x^2√(42-6x^2)-(95/2)*1/6^2∫√(42-6x^2)d(42-6x^2)
=-(95/12)x^2√(42-6x^2)-(95/3)*1/6^2√(42-6x^2)^3+c
解法三：
思路：利用三角函數的代換關系，進行三角還原積分求得。
設x=√7sint，則cost=(1/√42)√(42-6x^2),此時：
I=(3990/6)*√7∫sin^3td[√7sint]/√(42-42sin^2t),
=95*7^2∫sin^3tcostdt/√42*cost,
=(3990√42/6^2)∫sin^3tdt,
=(3990√42/6^2)∫sint(1-cos^2 t)dt
=(3990√42/6^2)∫sintdt-(3990√42/6^2)∫sintcos^2 tdt
=-(3990√42/6^2)cost+(3990√42/6^2)∫cos^2tdcost
=-(3990√42 /6^2)cost+(3990√42/3*6^2)cos^3t+c
=-(3990/6^2)√(42-6x^2)+(95/3)*(1/6^2)√(42-6x^2)^3+c.
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