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^4)/(x^2+a^2)

解答

\int \frac{x ^{4}}{x ^{2} + a ^{2}} d x

解答

\frac{x ^{3}}{3} - a^{2} x + a^{3} arctan (\frac{x}{a}) + C

求解步骤

\int \frac{x ^{4}}{x ^{2} + a ^{2}} d x

使用换元积分法

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

提出常数:

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

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

长除

\frac{u ^{4}}{u ^{2} + 1} : u^{2} - 1 + \frac{1}{u ^{2} + 1}

= a^{3} \cdot \int u^{2} - 1 + \frac{1}{u ^{2} + 1} 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

= a^{3} (\int u^{2} d u - \int 1 d u + \int \frac{1}{u ^{2} + 1} d u)

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

\int 1 d u = u

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

= a^{3} (\frac{u ^{3}}{3} - u + arctan (u))

u = \frac{x}{a}

代回

= a^{3} (\frac{( \frac{x}{a} ) ^{3}}{3} - \frac{x}{a} + arctan (\frac{x}{a}))

化简

a^{3} (\frac{( \frac{x}{a} ) ^{3}}{3} - \frac{x}{a} + arctan (\frac{x}{a})) : \frac{x ^{3}}{3} - a^{2} x + a^{3} arctan (\frac{x}{a})

= \frac{x ^{3}}{3} - a^{2} x + a^{3} arctan (\frac{x}{a})

解答补常数

= \frac{x ^{3}}{3} - a^{2} x + a^{3} arctan (\frac{x}{a}) + C

作图

流行的例子

integral of 4x^3(2x^4+4)^7

\int 4 x^{3} (2 x^{4} + 4)^{7} d x

integral of (x^3+5x^2-2)

\int (x^{3} + 5 x^{2} - 2) d x

integral of ((x^2+x+6))/(x^3+4x^2+6x)

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

integral of 2x*sqrt(1+x^2)

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

integral of (sin(2x))/((1-cos(2x))^3)

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