What is the integral of (z^2)/(sqrt(p^2+z^2)) ?

The integral of (z^2)/(sqrt(p^2+z^2)) is 1/2 zsqrt(z^2+p^2)-(p^2)/2 ln((|z+sqrt(z^2+p^2)|)/(|p|))+C

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integral of (z^2)/(sqrt(p^2+z^2))

\int \frac{z ^{2}}{p ^{2} + z ^{2}} d z

\frac{1}{2} z z^{2} + p^{2} - \frac{p ^{2}}{2} ln \frac{z + z ^{2} + p ^{2}}{∣ p ∣} + C

Solution steps

\int \frac{z ^{2}}{p ^{2} + z ^{2}} d z

Apply Trigonometric Substitution

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

= \int p^{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

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

Rewrite using trig identities

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

Expand

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

= p^{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

= p^{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) ∣

= p^{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}{p} z)

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

Simplify

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

= \frac{1}{2} z z^{2} + p^{2} - \frac{p ^{2}}{2} ln \frac{z + z ^{2} + p ^{2}}{∣ p ∣}

Add a constant to the solution

= \frac{1}{2} z z^{2} + p^{2} - \frac{p ^{2}}{2} ln \frac{z + z ^{2} + p ^{2}}{∣ p ∣} + C

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

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f (x) = (x - 3) (4 x + 2)

f (x) = 2 sin^{2} (3 x) + 2 cos^{2} (3 x)

\int_{- 2}^{2} x + 2 d x

What is the integral of (z^2)/(sqrt(p^2+z^2)) ?
The integral of (z^2)/(sqrt(p^2+z^2)) is 1/2 zsqrt(z^2+p^2)-(p^2)/2 ln((|z+sqrt(z^2+p^2)|)/(|p|))+C