What is the integral of (r^2)/(sqrt(x^2+r^2)) ?

The integral of (r^2)/(sqrt(x^2+r^2)) is 1/2 rsqrt(r^2+x^2)-(x^2)/2 ln((|r+sqrt(r^2+x^2)|)/(|x|))+C

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integral of (r^2)/(sqrt(x^2+r^2))

\int \frac{r ^{2}}{x ^{2} + r ^{2}} d r

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

Solution steps

\int \frac{r ^{2}}{x ^{2} + r ^{2}} d r

Apply Trigonometric Substitution

: \int x^{2} tan^{2} (u) sec (u) d u

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

Take the constant out:

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

= x^{2} \cdot \int tan^{2} (u) sec (u) d u

Rewrite using trig identities

= x^{2} \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

Expand

(- 1 + sec^{2} (u)) sec (u) : - sec (u) + sec^{3} (u)

= x^{2} \cdot \int - sec (u) + sec^{3} (u) 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

= x^{2} (- \int sec (u) d u + \int sec^{3} (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= x^{2} (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arctan (\frac{1}{x} r)

= x^{2} (- ln tan (arctan (\frac{1}{x} r)) + sec (arctan (\frac{1}{x} r)) + \frac{1}{2} sec (arctan (\frac{1}{x} r)) tan (arctan (\frac{1}{x} r)) + \frac{1}{2} ln tan (arctan (\frac{1}{x} r)) + sec (arctan (\frac{1}{x} r)))

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (r^2)/(sqrt(x^2+r^2)) ?
The integral of (r^2)/(sqrt(x^2+r^2)) is 1/2 rsqrt(r^2+x^2)-(x^2)/2 ln((|r+sqrt(r^2+x^2)|)/(|x|))+C