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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

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

= \int y^{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

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

Rewrite using trig identities

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

Expand

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

= y^{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

= y^{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) ∣

= y^{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}{y} x)

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

Simplify

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

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

Add a constant to the solution

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

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

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