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受欢迎的微积分 >

integral of 1/(x^2sqrt(16-x^2))

解答

\int \frac{1}{x ^{2} 16 - x ^{2}} d x

解答

- \frac{16 - x ^{2}}{16 x} + C

求解步骤

\int \frac{1}{x ^{2} 16 - x ^{2}} d x

使用三角换元法

= \int \frac{cot ( u )}{16 cos ( u ) sin ( u )} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{16} \cdot \int \frac{cot ( u )}{cos ( u ) sin ( u )} d u

化简

\frac{cot ( u )}{cos ( u ) sin ( u )} : csc^{2} (u)

= \frac{1}{16} \cdot \int csc^{2} (u) d u

使用积分常见定则:

\int csc^{2} (u) d u = - cot (u)

= \frac{1}{16} (- cot (u))

u = arcsin (\frac{1}{4} x)

代回

= \frac{1}{16} (- cot (arcsin (\frac{1}{4} x)))

化简

\frac{1}{16} (- cot (arcsin (\frac{1}{4} x))) : - \frac{16 - x ^{2}}{16 x}

= - \frac{16 - x ^{2}}{16 x}

解答补常数

= - \frac{16 - x ^{2}}{16 x} + C

作图

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