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受欢迎的微积分 >

integral of 5/(x^2sqrt(x^2-9))

解答

\int \frac{5}{x ^{2} x ^{2} - 9} d x

解答

\frac{5 x ^{2} - 9}{9 x} + C

求解步骤

\int \frac{5}{x ^{2} x ^{2} - 9} d x

提出常数:

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

= 5 \cdot \int \frac{1}{x ^{2} x ^{2} - 9} d x

使用三角换元法

= 5 \cdot \int \frac{1}{9 sec ( u )} d u

提出常数:

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

= 5 \cdot \frac{1}{9} \cdot \int \frac{1}{sec ( u )} d u

使用三角恒等式改写

= 5 \cdot \frac{1}{9} \cdot \int cos (u) d u

使用积分常见定则:

\int cos (u) d u = sin (u)

= 5 \cdot \frac{1}{9} sin (u)

u = arcsec (\frac{1}{3} x)

代回

= 5 \cdot \frac{1}{9} sin (arcsec (\frac{1}{3} x))

化简

5 \cdot \frac{1}{9} sin (arcsec (\frac{1}{3} x)) : \frac{5 x ^{2} - 9}{9 x}

= \frac{5 x ^{2} - 9}{9 x}

解答补常数

= \frac{5 x ^{2} - 9}{9 x} + C

作图

流行的例子

(d^2)/(dx^2)(3xsin(x^2))

\frac{d ^{2}}{d x ^{2}} (3 x sin (x^{2}))

integral of x/(x-9)

\int \frac{x}{x - 9} d x

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d er i v a t i v e sin^{5} (x^{3})

limit as x approaches 5 of |x-5|

x \to 5 lim (∣ x - 5 ∣)

integral of-2/(sqrt(100-9t^2))

\int - \frac{2}{100 - 9 t ^{2}} d t