What is the integral of (x^3)/((x^2-256)^{3/2)} ?

The integral of (x^3)/((x^2-256)^{3/2)} is (x^2-512)/((x^2-256)^{1/2)}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (x^3)/((x^2-256)^{3/2)}

\int \frac{x ^{3}}{( x ^{2} - 256 ) ^{\frac{3}{2}}} d x

\frac{x ^{2} - 512}{( x ^{2} - 256 ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{x ^{3}}{( x ^{2} - 256 ) ^{\frac{3}{2}}} d x

Apply u-substitution

= \int \frac{u + 256}{2 u ^{\frac{3}{2}}} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{u + 256}{u ^{\frac{3}{2}}} d u

Expand

\frac{u + 256}{u ^{\frac{3}{2}}} : \frac{1}{u ^{\frac{1}{2}}} + \frac{256}{u ^{\frac{3}{2}}}

= \frac{1}{2} \cdot \int \frac{1}{u ^{\frac{1}{2}}} + \frac{256}{u ^{\frac{3}{2}}} 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}{2} (\int \frac{1}{u ^{\frac{1}{2}}} d u + \int \frac{256}{u ^{\frac{3}{2}}} d u)

\int \frac{1}{u ^{\frac{1}{2}}} d u = 2 u^{\frac{1}{2}}

\int \frac{256}{u ^{\frac{3}{2}}} d u = - \frac{512}{u ^{\frac{1}{2}}}

= \frac{1}{2} (2 u^{\frac{1}{2}} - \frac{512}{u ^{\frac{1}{2}}})

Substitute back

u = x^{2} - 256

= \frac{1}{2} (2 (x^{2} - 256)^{\frac{1}{2}} - \frac{512}{( x ^{2} - 256 ) ^{\frac{1}{2}}})

Simplify

\frac{1}{2} (2 (x^{2} - 256)^{\frac{1}{2}} - \frac{512}{( x ^{2} - 256 ) ^{\frac{1}{2}}}) : \frac{x ^{2} - 512}{( x ^{2} - 256 ) ^{\frac{1}{2}}}

= \frac{x ^{2} - 512}{( x ^{2} - 256 ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x ^{2} - 512}{( x ^{2} - 256 ) ^{\frac{1}{2}}} + C

s l o p e (0, - 10), (6, 10)

t a y l or \frac{1}{x}, at 1

\int e^{iπ z} d z

d er i v a t i v e f (t) = t - \frac{1}{t}

\int \frac{sec ( x ) tan ( x )}{sec ( x ) - 2} d x

What is the integral of (x^3)/((x^2-256)^{3/2)} ?
The integral of (x^3)/((x^2-256)^{3/2)} is (x^2-512)/((x^2-256)^{1/2)}+C