zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of (21)/(4+25r^2)

解答

\int \frac{21}{4 + 25 r ^{2}} d r

解答

\frac{21}{10} arctan (\frac{5}{2} r) + C

求解步骤

\int \frac{21}{4 + 25 r ^{2}} d r

提出常数:

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

= 21 \cdot \int \frac{1}{4 + 25 r ^{2}} d r

使用换元积分法

= 21 \cdot \int \frac{1}{10 ( u ^{2} + 1 )} d u

提出常数:

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

= 21 \cdot \frac{1}{10} \cdot \int \frac{1}{u ^{2} + 1} d u

使用积分常见定则:

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

= 21 \cdot \frac{1}{10} arctan (u)

u = \frac{5}{2} r

代回

= 21 \cdot \frac{1}{10} arctan (\frac{5}{2} r)

化简

21 \cdot \frac{1}{10} arctan (\frac{5}{2} r) : \frac{21}{10} arctan (\frac{5}{2} r)

= \frac{21}{10} arctan (\frac{5}{2} r)

解答补常数

= \frac{21}{10} arctan (\frac{5}{2} r) + C

作图

流行的例子

derivative (-1)/((x-1)^2)

d er i v a t i v e \frac{- 1}{( x - 1 ) ^{2}}

derivative of (x^4-2/(x^2-2))

\frac{d}{d x} (\frac{x ^{4} - 2}{x ^{2} - 2})

(\partial)/(\partial x)(6x^2y+6xy^3)

\frac{\partial}{\partial x} (6 x^{2} y + 6 x y^{3})

limit as x approaches-5 of g(x)

x \to - 5 lim (g (x))

d/(d{x)}(({x})/(sqrt({x)^2+{y)^2+{z}^2}})

\frac{d}{d x} (\frac{x}{x ^{2} + y ^{2} + z ^{2}})