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 sqrt(36x^2+12x+10)

解答

\int 36 x^{2} + 12 x + 10 d x

解答

\frac{1}{12} (6 x 36 x^{2} + 12 x + 10 + 36 x^{2} + 12 x + 10 + 9 ln (\frac{6 ( x + \frac{1}{6} ) + 36 x ^{2} + 12 x + 10}{3})) + C

求解步骤

\int 36 x^{2} + 12 x + 10 d x

36 x^{2} + 12 x + 10 = 2 18 x^{2} + 6 x + 5

= \int 2 18 x^{2} + 6 x + 5 d x

提出常数:

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

= 2 \cdot \int 18 x^{2} + 6 x + 5 d x

对

18 x^{2} + 6 x + 5

配方:

18 (x + \frac{1}{6})^{2} + \frac{9}{2}

= 2 \cdot \int 18 (x + \frac{1}{6})^{2} + \frac{9}{2} d x

使用换元积分法

= 2 \cdot \int \frac{36 u ^{2} + 9}{2} d u

\frac{36 u ^{2} + 9}{2} = \frac{9}{2} 4 u^{2} + 1

= 2 \cdot \int \frac{9}{2} 4 u^{2} + 1 d u

提出常数:

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

= 2 \frac{9}{2} \cdot \int 4 u^{2} + 1 d u

\frac{9}{2} = \frac{3}{2}

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

使用三角换元法

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

提出常数:

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

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

使用降次积分法

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

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

= 2 \frac{3}{2} \cdot \frac{1}{2} (\frac{sec ^{2} ( v ) sin ( v )}{2} + \frac{1}{2} ln ∣ tan (v) + sec (v) ∣)

代回

= 2 \frac{3}{2} \cdot \frac{1}{2} (\frac{sec ^{2} ( arctan ( 2 ( x + \frac{1}{6} ) ) ) sin ( arctan ( 2 ( x + \frac{1}{6} ) ) )}{2} + \frac{1}{2} ln tan (arctan (2 (x + \frac{1}{6}))) + sec (arctan (2 (x + \frac{1}{6}))))

化简

2 \frac{3}{2} \cdot \frac{1}{2} (\frac{sec ^{2} ( arctan ( 2 ( x + \frac{1}{6} ) ) ) sin ( arctan ( 2 ( x + \frac{1}{6} ) ) )}{2} + \frac{1}{2} ln tan (arctan (2 (x + \frac{1}{6}))) + sec (arctan (2 (x + \frac{1}{6})))) : \frac{1}{12} (6 x 36 x^{2} + 12 x + 10 + 36 x^{2} + 12 x + 10 + 9 ln (\frac{6 ( x + \frac{1}{6} ) + 36 x ^{2} + 12 x + 10}{3}))

= \frac{1}{12} (6 x 36 x^{2} + 12 x + 10 + 36 x^{2} + 12 x + 10 + 9 ln (\frac{6 ( x + \frac{1}{6} ) + 36 x ^{2} + 12 x + 10}{3}))

解答补常数

= \frac{1}{12} (6 x 36 x^{2} + 12 x + 10 + 36 x^{2} + 12 x + 10 + 9 ln (\frac{6 ( x + \frac{1}{6} ) + 36 x ^{2} + 12 x + 10}{3})) + C

作图

流行的例子

integral of sqrt(3x)-x

\int 3 x - x d x

integral of (7x-5)/(x^2-x-2)

\int \frac{7 x - 5}{x ^{2} - x - 2} d x

integral of x^35^x

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

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

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

integral of cos^7(x)sin^4(x)

\int cos^{7} (x) sin^{4} (x) d x