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)/((x^2+1)^{3/2)}

解答

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

解答

ln x + 1 + x^{2} - \frac{x}{( 1 + x ^{2} ) ^{\frac{1}{2}}} + C

求解步骤

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

使用三角换元法

= \int tan^{2} (u) cos (u) d u

化简

tan^{2} (u) cos (u) : sin (u) tan (u)

= \int sin (u) tan (u) d u

使用三角恒等式改写

= \int \frac{1 - cos ^{2} ( u )}{cos ( u )} d u

乘开

\frac{1 - cos ^{2} ( u )}{cos ( u )} : \frac{1}{cos ( u )} - cos (u)

= \int \frac{1}{cos ( u )} - cos (u) 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

= \int \frac{1}{cos ( u )} d u - \int cos (u) d u

\int \frac{1}{cos ( u )} d u = ln ∣ tan (u) + sec (u) ∣

\int cos (u) d u = sin (u)

= ln ∣ tan (u) + sec (u) ∣ - sin (u)

u = arctan (x)

代回

= ln ∣ tan (arctan (x)) + sec (arctan (x)) ∣ - sin (arctan (x))

化简

ln ∣ tan (arctan (x)) + sec (arctan (x)) ∣ - sin (arctan (x)) : ln x + 1 + x^{2} - \frac{x}{( 1 + x ^{2} ) ^{\frac{1}{2}}}

= ln x + 1 + x^{2} - \frac{x}{( 1 + x ^{2} ) ^{\frac{1}{2}}}

解答补常数

= ln x + 1 + x^{2} - \frac{x}{( 1 + x ^{2} ) ^{\frac{1}{2}}} + C

作图

流行的例子

\frac{d y}{d x} = - \frac{2}{x}

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

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

integral of 3/(x^2sqrt(1+x^2))

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

(\partial)/(\partial y)(sqrt(x^2+y^2))

\frac{\partial}{\partial y} (x^{2} + y^{2})

integral from 0 to 8 of 5/(x^{2/3)}

\int_{0}^{8} \frac{5}{x ^{\frac{2}{3}}} d x