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 3/(x^2+4x+14)

解答

\int \frac{3}{x ^{2} + 4 x + 14} d x

解答

\frac{3}{10} arctan (\frac{1}{10} (x + 2)) + C

求解步骤

\int \frac{3}{x ^{2} + 4 x + 14} d x

提出常数:

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

= 3 \cdot \int \frac{1}{x ^{2} + 4 x + 14} d x

对

x^{2} + 4 x + 14

配方:

(x + 2)^{2} + 10

= 3 \cdot \int \frac{1}{( x + 2 ) ^{2} + 10} d x

使用换元积分法

= 3 \cdot \int \frac{1}{u ^{2} + 10} d u

使用换元积分法

= 3 \cdot \int \frac{1}{10 ( v ^{2} + 1 )} d v

提出常数:

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

= 3 \cdot \frac{1}{10} \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)

= 3 \cdot \frac{1}{10} arctan (v)

代回

= 3 \cdot \frac{1}{10} arctan (\frac{x + 2}{10})

化简

3 \cdot \frac{1}{10} arctan (\frac{x + 2}{10}) : \frac{3}{10} arctan (\frac{1}{10} (x + 2))

= \frac{3}{10} arctan (\frac{1}{10} (x + 2))

解答补常数

= \frac{3}{10} arctan (\frac{1}{10} (x + 2)) + C

作图

流行的例子

integral of sin(x)-12cos(x)

\int sin (x) - 12 cos (x) d x

integral of cot(3x)+csc(3x)

\int cot (3 x) + csc (3 x) d x

integral of 1/((x^2+3/4)^2)

\int \frac{1}{( x ^{2} + \frac{3}{4} ) ^{2}} d x

integral of (e^{2x})/(e^{x+3)}

\int \frac{e ^{2 x}}{e ^{x + 3}} d x

integral of x^2\sqrt[4]{x^3+1}

\int x^{2} 4 x^{3} + 1 d x