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

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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

= \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

\frac{d}{d x} (2 π \frac{0.5}{9.8})

2 x y \frac{d y}{d x} = 1

f (x) = x^{3} (x - 4)^{2}

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