What is the integral of 1/(z^3sqrt(z^2-25)) ?

The integral of 1/(z^3sqrt(z^2-25)) is 1/250 (arcsec(1/5 z)+1/2 sin(2arcsec(1/5 z)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(z^3sqrt(z^2-25))

\int \frac{1}{z ^{3} z ^{2} - 25} d z

\frac{1}{250} (arcsec (\frac{1}{5} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{5} z))) + C

Solution steps

\int \frac{1}{z ^{3} z ^{2} - 25} d z

Apply Trigonometric Substitution

: \int \frac{1}{125 sec ^{2} ( u )} d u

= \int \frac{1}{125 sec ^{2} ( u )} d u

Take the constant out:

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

= \frac{1}{125} \cdot \int \frac{1}{sec ^{2} ( u )} d u

Rewrite using trig identities

= \frac{1}{125} \cdot \int \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= \frac{1}{125} \cdot \frac{1}{2} \cdot \int 1 + cos (2 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

= \frac{1}{125} \cdot \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= \frac{1}{125} \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = arcsec (\frac{1}{5} z)

= \frac{1}{125} \cdot \frac{1}{2} (arcsec (\frac{1}{5} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{5} z)))

Simplify

\frac{1}{125} \cdot \frac{1}{2} (arcsec (\frac{1}{5} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{5} z))) : \frac{1}{250} (arcsec (\frac{1}{5} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{5} z)))

= \frac{1}{250} (arcsec (\frac{1}{5} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{5} z)))

Add a constant to the solution

= \frac{1}{250} (arcsec (\frac{1}{5} z) + \frac{1}{2} sin (2 arcsec (\frac{1}{5} z))) + C

\int (cos^{2} (2 x))^{2} d x

\frac{d}{d x} (2 \cdot sin (x))

\int 18 sin^{2} (x) cos^{2} (x) d x

\frac{( 2 s + 2 )}{( s ^{2} + 13 s + 12 )}

\int - \frac{4 x}{( 1 - 2 x ^{2} ) ^{3}} d x

What is the integral of 1/(z^3sqrt(z^2-25)) ?
The integral of 1/(z^3sqrt(z^2-25)) is 1/250 (arcsec(1/5 z)+1/2 sin(2arcsec(1/5 z)))+C