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

解答

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

解答

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

求解步骤

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

(x^{2} + 1)^{3} = (x^{2} + 1)^{\frac{3}{2}},

假定

(x^{2} + 1) \geq 0

= \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}}

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

解答补常数

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

作图

流行的例子

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

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

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

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

integral of sin(e^{3x})

\int sin (e^{3 x}) d x

integral of 1/((e^x-e^{-x))}

\int \frac{1}{( e ^{x} - e ^{- x} )} d x

integral of sqrt(1+x^2)(x^5)

\int 1 + x^{2} (x^{5}) d x