integral of 5/(usqrt(u^2-1))

解答

\int \frac{5}{u u ^{2} - 1} d u

5 arctan (u^{2} - 1) + C

求解步骤

\int \frac{5}{u u ^{2} - 1} d u

提出常数:

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

= 5 \cdot \int \frac{1}{u u ^{2} - 1} d u

使用换元积分法

= 5 \cdot \int \frac{1}{v ^{2} + 1} d v

使用积分常见定则:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= 5 arctan (v)

v = u^{2} - 1

代回

= 5 arctan (u^{2} - 1)

解答补常数

= 5 arctan (u^{2} - 1) + C

d er i v a t i v e f (t) = \frac{9}{t}

\frac{d}{d x} (\frac{x - 5}{x ^{2} + 1})

\int sin (8 x - 2) d x

f^{^{'}} (x) = 6 x^{2} - 8 x, f (3) = 27

x \to - 9 lim (\frac{- 3 x}{x + 9})