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 (5x^2)/((36+x^2)^2)

解答

\int \frac{5 x ^{2}}{( 36 + x ^{2} ) ^{2}} d x

解答

\frac{5 ( x ^{2} arctan ( \frac{x}{6} ) - 6 x + 36 arctan ( \frac{x}{6} ) )}{12 ( 36 + x ^{2} )} + C

求解步骤

\int \frac{5 x ^{2}}{( 36 + x ^{2} ) ^{2}} d x

提出常数:

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

= 5 \cdot \int \frac{x ^{2}}{( 36 + x ^{2} ) ^{2}} d x

使用换元积分法

= 5 \cdot \int \frac{u ^{2}}{6 ( u ^{2} + 1 ) ^{2}} d u

提出常数:

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

= 5 \cdot \frac{1}{6} \cdot \int \frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} d u

将

\frac{u ^{2}}{( u ^{2} + 1 ) ^{2}}

用部份分式展开:

\frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}}

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

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

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

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

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

= 5 \cdot \frac{1}{6} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

u = \frac{x}{6}

代回

= 5 \cdot \frac{1}{6} (arctan (\frac{x}{6}) - \frac{1}{2} (arctan (\frac{x}{6}) + \frac{\frac{x}{6}}{1 + ( \frac{x}{6} ) ^{2}}))

化简

5 \cdot \frac{1}{6} (arctan (\frac{x}{6}) - \frac{1}{2} (arctan (\frac{x}{6}) + \frac{\frac{x}{6}}{1 + ( \frac{x}{6} ) ^{2}})) : \frac{5 ( x ^{2} arctan ( \frac{x}{6} ) - 6 x + 36 arctan ( \frac{x}{6} ) )}{12 ( 36 + x ^{2} )}

= \frac{5 ( x ^{2} arctan ( \frac{x}{6} ) - 6 x + 36 arctan ( \frac{x}{6} ) )}{12 ( 36 + x ^{2} )}

解答补常数

= \frac{5 ( x ^{2} arctan ( \frac{x}{6} ) - 6 x + 36 arctan ( \frac{x}{6} ) )}{12 ( 36 + x ^{2} )} + C

作图

流行的例子

integral of (4x^{1/3}+1)

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

integral of 2/3 (x+2)^{3/2}

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

integral of 2/(2x^2+1)

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

integral of e^{-1/2 y}

\int e^{- \frac{1}{2} y} d y

integral of 3xe^{x^3-1}

\int 3 x e^{x^{3} - 1} d x