What is the integral of (x^2)/(sqrt(a^2+(x-b)^2)) ?

The integral of (x^2)/(sqrt(a^2+(x-b)^2)) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x^2)/(sqrt(a^2+(x-b)^2))

\int \frac{x ^{2}}{a ^{2} + ( x - b ) ^{2}} d x

\frac{1}{2} x a^{2} + (x - b)^{2} + b^{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + \frac{3}{2} b a^{2} + (x - b)^{2} - \frac{a ^{2}}{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + C

Solution steps

\int \frac{x ^{2}}{a ^{2} + ( x - b ) ^{2}} d x

Apply u-substitution

= \int \frac{( u + b ) ^{2}}{a ^{2} + u ^{2}} d u

Expand

\frac{( u + b ) ^{2}}{a ^{2} + u ^{2}} : \frac{u ^{2}}{a ^{2} + u ^{2}} + \frac{2 b u}{a ^{2} + u ^{2}} + \frac{b ^{2}}{a ^{2} + u ^{2}}

= \int \frac{u ^{2}}{a ^{2} + u ^{2}} + \frac{2 b u}{a ^{2} + u ^{2}} + \frac{b ^{2}}{a ^{2} + u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{u ^{2}}{a ^{2} + u ^{2}} d u + \int \frac{2 b u}{a ^{2} + u ^{2}} d u + \int \frac{b ^{2}}{a ^{2} + u ^{2}} d u

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

\int \frac{2 b u}{a ^{2} + u ^{2}} d u = 2 b a^{2} + u^{2}

\int \frac{b ^{2}}{a ^{2} + u ^{2}} d u = b^{2} ln (\frac{u + a ^{2} + u ^{2}}{∣ a ∣})

= \frac{1}{2} u u^{2} + a^{2} - \frac{a ^{2}}{2} ln (\frac{u + u ^{2} + a ^{2}}{∣ a ∣}) + 2 b a^{2} + u^{2} + b^{2} ln (\frac{u + a ^{2} + u ^{2}}{∣ a ∣})

Substitute back

u = x - b

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

Simplify

\frac{1}{2} (x - b) (x - b)^{2} + a^{2} - \frac{a ^{2}}{2} ln \frac{x - b + ( x - b ) ^{2} + a ^{2}}{∣ a ∣} + 2 b a^{2} + (x - b)^{2} + b^{2} ln \frac{x - b + a ^{2} + ( x - b ) ^{2}}{∣ a ∣} : \frac{1}{2} x a^{2} + (x - b)^{2} + b^{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + \frac{3}{2} b a^{2} + (x - b)^{2} - \frac{a ^{2}}{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣}

= \frac{1}{2} x a^{2} + (x - b)^{2} + b^{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + \frac{3}{2} b a^{2} + (x - b)^{2} - \frac{a ^{2}}{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣}

Add a constant to the solution

= \frac{1}{2} x a^{2} + (x - b)^{2} + b^{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + \frac{3}{2} b a^{2} + (x - b)^{2} - \frac{a ^{2}}{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + C

\int 5 x cos (8 x) d x

d er i v a t i v e f (x) = x^{3} (x - 4)^{2}

\frac{d}{d x} (- 2 x (e^{- 2 x}))

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

\int \frac{x + 1}{2 x ^{2} + 4 x + 5} d x

What is the integral of (x^2)/(sqrt(a^2+(x-b)^2)) ?
The integral of (x^2)/(sqrt(a^2+(x-b)^2)) is