What is the integral of 1/(x^3sqrt(x^2-64)) ?

The integral of 1/(x^3sqrt(x^2-64)) is 1/1024 (arcsec(1/8 x)+1/2 sin(2arcsec(1/8 x)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(x^3sqrt(x^2-64))

\int \frac{1}{x ^{3} x ^{2} - 64} d x

\frac{1}{1024} (arcsec (\frac{1}{8} x) + \frac{1}{2} sin (2 arcsec (\frac{1}{8} x))) + C

Solution steps

\int \frac{1}{x ^{3} x ^{2} - 64} d x

Apply Trigonometric Substitution

: \int \frac{1}{512 sec ^{2} ( u )} d u

= \int \frac{1}{512 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}{512} \cdot \int \frac{1}{sec ^{2} ( u )} d u

Rewrite using trig identities

= \frac{1}{512} \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}{512} \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}{512} \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}{512} \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = arcsec (\frac{1}{8} x)

= \frac{1}{512} \cdot \frac{1}{2} (arcsec (\frac{1}{8} x) + \frac{1}{2} sin (2 arcsec (\frac{1}{8} x)))

Simplify

\frac{1}{512} \cdot \frac{1}{2} (arcsec (\frac{1}{8} x) + \frac{1}{2} sin (2 arcsec (\frac{1}{8} x))) : \frac{1}{1024} (arcsec (\frac{1}{8} x) + \frac{1}{2} sin (2 arcsec (\frac{1}{8} x)))

= \frac{1}{1024} (arcsec (\frac{1}{8} x) + \frac{1}{2} sin (2 arcsec (\frac{1}{8} x)))

Add a constant to the solution

= \frac{1}{1024} (arcsec (\frac{1}{8} x) + \frac{1}{2} sin (2 arcsec (\frac{1}{8} x))) + C

\int 9 x e^{5 x} d x

6 x, 0, 2

x^{2} + 2 x + 1, - 3, 0

f (x) = - 4 x^{2} - 3 x + 2, at x = - 1

\int_{1}^{3} x^{2} \frac{2.25}{x ^{3}} d x

What is the integral of 1/(x^3sqrt(x^2-64)) ?
The integral of 1/(x^3sqrt(x^2-64)) is 1/1024 (arcsec(1/8 x)+1/2 sin(2arcsec(1/8 x)))+C