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

解答

\int \frac{( x + 2 ) ^{5}}{6 + 4 x + x ^{2}} d x

解答

42 \frac{1}{2} (x^{2} + 4 x + 6) - \frac{4}{3} (x^{2} + 4 x + 6)^{\frac{3}{2}} + \frac{1}{5} (x^{2} + 4 x + 6)^{\frac{5}{2}} + C

求解步骤

\int \frac{( x + 2 ) ^{5}}{6 + 4 x + x ^{2}} d x

使用换元积分法

= \int \frac{u ^{5}}{u ^{2} + 2} d u

使用三角换元法

= \int 42 tan^{5} (v) sec (v) d v

提出常数:

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

= 42 \cdot \int tan^{5} (v) sec (v) d v

使用三角恒等式改写

= 42 \cdot \int (- 1 + sec^{2} (v))^{2} tan (v) sec (v) d v

使用换元积分法

= 42 \cdot \int (- 1 + w^{2})^{2} d w

乘开

(- 1 + w^{2})^{2} : 1 - 2 w^{2} + w^{4}

= 42 \cdot \int 1 - 2 w^{2} + w^{4} 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

= 42 (\int 1 d w - \int 2 w^{2} d w + \int w^{4} d w)

\int 1 d w = w

\int 2 w^{2} d w = \frac{2 w ^{3}}{3}

\int w^{4} d w = \frac{w ^{5}}{5}

= 42 (w - \frac{2 w ^{3}}{3} + \frac{w ^{5}}{5})

代回

= 42 sec (arctan (\frac{1}{2} (x + 2))) - \frac{2 sec ^{3} ( arctan ( \frac{1}{2} ( x + 2 ) ) )}{3} + \frac{sec ^{5} ( arctan ( \frac{1}{2} ( x + 2 ) ) )}{5}

化简

42 sec (arctan (\frac{1}{2} (x + 2))) - \frac{2 sec ^{3} ( arctan ( \frac{1}{2} ( x + 2 ) ) )}{3} + \frac{sec ^{5} ( arctan ( \frac{1}{2} ( x + 2 ) ) )}{5} : 42 \frac{1}{2} (x^{2} + 4 x + 6) - \frac{4}{3} (x^{2} + 4 x + 6)^{\frac{3}{2}} + \frac{1}{5} (x^{2} + 4 x + 6)^{\frac{5}{2}}

= 42 \frac{1}{2} (x^{2} + 4 x + 6) - \frac{4}{3} (x^{2} + 4 x + 6)^{\frac{3}{2}} + \frac{1}{5} (x^{2} + 4 x + 6)^{\frac{5}{2}}

解答补常数

= 42 \frac{1}{2} (x^{2} + 4 x + 6) - \frac{4}{3} (x^{2} + 4 x + 6)^{\frac{3}{2}} + \frac{1}{5} (x^{2} + 4 x + 6)^{\frac{5}{2}} + C

作图

流行的例子

integral of 1/(\frac{x^2){2}+1}

\int \frac{1}{\frac{x ^{2}}{2} + 1} d x

integral of (e^x)/((e^{2x)-1)}

\int \frac{e ^{x}}{( e ^{2 x} - 1 )} d x

integral of (4x-1)/(sqrt(4x^2-9))

\int \frac{4 x - 1}{4 x ^{2} - 9} d x

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

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

integral of x(x^2+6)^9

\int x (x^{2} + 6)^{9} d x