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 5/(sqrt(6+8x-4x^2))

解答

\int \frac{5}{6 + 8 x - 4 x ^{2}} d x

解答

\frac{5}{2} arcsin (\frac{2}{5} (x - 1)) + C

求解步骤

\int \frac{5}{6 + 8 x - 4 x ^{2}} 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}{6 + 8 x - 4 x ^{2}} d x

对

6 + 8 x - 4 x^{2}

配方:

- 4 (x - 1)^{2} + 10

= 5 \cdot \int \frac{1}{- 4 ( x - 1 ) ^{2} + 10} d x

使用换元积分法

= 5 \cdot \int \frac{1}{- 4 u ^{2} + 10} d u

使用三角换元法

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

提出常数:

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

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

常数积分:

\int a d x = a x

= 5 \cdot \frac{1}{2} \cdot 1 \cdot v

代回

= 5 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (\frac{2}{10} (x - 1))

化简

5 \cdot \frac{1}{2} \cdot 1 \cdot arcsin (\frac{2}{10} (x - 1)) : \frac{5}{2} arcsin (\frac{2}{5} (x - 1))

= \frac{5}{2} arcsin (\frac{2}{5} (x - 1))

解答补常数

= \frac{5}{2} arcsin (\frac{2}{5} (x - 1)) + C

作图

流行的例子

integral of sqrt(9+7x^2)

\int 9 + 7 x^{2} d x

integral of (1+3x^2)(5-x)

\int (1 + 3 x^{2}) (5 - x) d x

integral of 1/(x(x+3)(x+1))

\int \frac{1}{x ( x + 3 ) ( x + 1 )} d x

integral of xcos(x^4)

\int x cos (x^{4}) d x

integral of (e^xcos(e^x))

\int (e^{x} cos (e^{x})) d x