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受欢迎的微积分 >

integral of 1/((16-t^2)^{3/2)}

解答

\int \frac{1}{( 16 - t ^{2} ) ^{\frac{3}{2}}} d t

解答

\frac{t}{16 ( 16 - t ^{2} ) ^{\frac{1}{2}}} + C

求解步骤

\int \frac{1}{( 16 - t ^{2} ) ^{\frac{3}{2}}} d t

使用三角换元法

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

使用积分常见定则:

\int \frac{1}{cos ^{2} ( u )} d u = tan (u)

= \frac{1}{16} tan (u)

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

代回

= \frac{1}{16} tan (arcsin (\frac{1}{4} t))

化简

\frac{1}{16} tan (arcsin (\frac{1}{4} t)) : \frac{t}{16 ( 16 - t ^{2} ) ^{\frac{1}{2}}}

= \frac{t}{16 ( 16 - t ^{2} ) ^{\frac{1}{2}}}

解答补常数

= \frac{t}{16 ( 16 - t ^{2} ) ^{\frac{1}{2}}} + C

作图

流行的例子

integral from-1 to 3 of 30-3y^2-9/2

\int_{- 1}^{3} 30 - 3 y^{2} - \frac{9}{2} d y

(dy)/(dx)= 1/(1+x^2)

\frac{d y}{d x} = \frac{1}{1 + x ^{2}}

limit as x approaches 0 of 5/(1-e^{1/x)}

x \to 0 lim (\frac{5}{1 - e ^{\frac{1}{x}}})

16 y^{^{''}} + 11 y = 0

integral of (e^{2x})/(e^x-1)

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