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/((x^2+6x+13)^2)

解答

\int \frac{1}{( x ^{2} + 6 x + 13 ) ^{2}} d x

解答

\frac{1}{16} (arctan (\frac{1}{2} (x + 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 3)))) + C

求解步骤

\int \frac{1}{( x ^{2} + 6 x + 13 ) ^{2}} d x

对

x^{2} + 6 x + 13

配方:

(x + 3)^{2} + 4

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

使用换元积分法

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

使用换元积分法

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

提出常数:

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

= \frac{1}{8} \cdot \int \frac{1}{( v ^{2} + 1 ) ^{2}} d v

使用三角换元法

= \frac{1}{8} \cdot \int \frac{1}{sec ^{4} ( w ) cos ^{2} ( w )} d w

使用三角恒等式改写

= \frac{1}{8} \cdot \int \frac{1 + cos ( 2 w )}{2} d w

提出常数:

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

= \frac{1}{8} \cdot \frac{1}{2} \cdot \int 1 + cos (2 w) d w

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{8} \cdot \frac{1}{2} (\int 1 d w + \int cos (2 w) d w)

\int 1 d w = w

\int cos (2 w) d w = \frac{1}{2} sin (2 w)

= \frac{1}{8} \cdot \frac{1}{2} (w + \frac{1}{2} sin (2 w))

代回

= \frac{1}{8} \cdot \frac{1}{2} (arctan (\frac{x + 3}{2}) + \frac{1}{2} sin (2 arctan (\frac{x + 3}{2})))

化简

\frac{1}{8} \cdot \frac{1}{2} (arctan (\frac{x + 3}{2}) + \frac{1}{2} sin (2 arctan (\frac{x + 3}{2}))) : \frac{1}{16} (arctan (\frac{1}{2} (x + 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 3))))

= \frac{1}{16} (arctan (\frac{1}{2} (x + 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 3))))

解答补常数

= \frac{1}{16} (arctan (\frac{1}{2} (x + 3)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 3)))) + C

作图

流行的例子

integral of (2+sqrt(x))/x

\int \frac{2 + x}{x} d x

integral of (5x+1)^3

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

integral of 4x^5-x^3

\int 4 x^{5} - x^{3} d x

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

\int \frac{1}{x ^{4} - 2 x ^{3}} d x

integral of 2/(3e^{-5x)-4}

\int \frac{2}{3 e ^{- 5 x} - 4} d x