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 (3x)/(sqrt(x^4+6x^2+5))

解答

\int \frac{3 x}{x ^{4} + 6 x ^{2} + 5} d x

解答

\frac{3}{2} ln \frac{1}{11} x^{2} + \frac{1}{11} (11 + x^{4}) + C

求解步骤

\int \frac{3 x}{x ^{4} + 6 x ^{2} + 5} d x

提出常数:

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

= 3 \cdot \int \frac{x}{x ^{4} + 6 x ^{2} + 5} d x

使用换元积分法

= 3 \cdot \int \frac{1}{4 u 6 u + u + 5} d u

提出常数:

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

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{u 6 u + u + 5} d u

使用指数法则:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{u 6 u + u + 5} = u^{- \frac{1}{2}} \frac{1}{6 + u + 5}

= 3 \cdot \frac{1}{4} \cdot \int u^{- \frac{1}{2}} \frac{1}{6 + u + 5} d u

化简

u^{- \frac{1}{2}} \frac{1}{6 + u + 5} : \frac{1}{u u + 11}

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{u u + 11} d u

使用换元积分法

= 3 \cdot \frac{1}{4} \cdot \int \frac{2}{v ^{2} + 11} d v

提出常数:

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

= 3 \cdot \frac{1}{4} \cdot 2 \cdot \int \frac{1}{v ^{2} + 11} d v

使用三角换元法

= 3 \cdot \frac{1}{4} \cdot 2 \cdot \int sec (w) d w

使用积分常见定则:

\int sec (w) d w = ln ∣ tan (w) + sec (w) ∣

= 3 \cdot \frac{1}{4} \cdot 2 ln ∣ tan (w) + sec (w) ∣

代回

= 3 \cdot \frac{1}{4} \cdot 2 ln tan (arctan (\frac{1}{11} x^{4})) + sec (arctan (\frac{1}{11} x^{4}))

化简

3 \cdot \frac{1}{4} \cdot 2 ln tan (arctan (\frac{1}{11} x^{4})) + sec (arctan (\frac{1}{11} x^{4})) : \frac{3}{2} ln \frac{1}{11} x^{2} + \frac{1}{11} (11 + x^{4})

= \frac{3}{2} ln \frac{1}{11} x^{2} + \frac{1}{11} (11 + x^{4})

解答补常数

= \frac{3}{2} ln \frac{1}{11} x^{2} + \frac{1}{11} (11 + x^{4}) + C

作图

流行的例子

derivative of (x^3-1/2)

\frac{d}{d x} (\frac{x ^{3} - 1}{2})

(\partial)/(\partial x)(2xcos(x))

\frac{\partial}{\partial x} (2 x cos (x))

(d^4)/(dx^4)(x^x)

\frac{d ^{4}}{d x ^{4}} (x^{x})

integral of e^{1/y}

\int e^{\frac{1}{y}} d y

derivative 4sqrt(x)-2x

d er i v a t i v e 4 x - 2 x