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 1/(2x^2-8x+16)

解答

\int \frac{1}{2 x ^{2} - 8 x + 16} d x

解答

\frac{1}{4} arctan (\frac{x - 2}{2}) + C

求解步骤

\int \frac{1}{2 x ^{2} - 8 x + 16} d x

对

2 x^{2} - 8 x + 16

配方:

2 (x - 2)^{2} + 8

= \int \frac{1}{2 ( x - 2 ) ^{2} + 8} d x

使用换元积分法

= \int \frac{1}{2 u ^{2} + 8} d u

使用换元积分法

= \int \frac{1}{4 ( v ^{2} + 1 )} d v

提出常数:

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

= \frac{1}{4} \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)

= \frac{1}{4} arctan (v)

代回

= \frac{1}{4} arctan (\frac{2}{2 2} (x - 2))

化简

\frac{1}{4} arctan (\frac{2}{2 2} (x - 2)) : \frac{1}{4} arctan (\frac{x - 2}{2})

= \frac{1}{4} arctan (\frac{x - 2}{2})

解答补常数

= \frac{1}{4} arctan (\frac{x - 2}{2}) + C

作图

流行的例子

integral of 6/(e^{2x)}

\int \frac{6}{e ^{2 x}} d x

integral of (3x^2+4)e^{x^3+4x}

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

integral of sec(t-1)tan(t-1)

\int sec (t - 1) tan (t - 1) d t

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

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

integral of (x+3)/(sqrt(9-x^2))

\int \frac{x + 3}{9 - x ^{2}} d x