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

The integral of x^2sqrt(x^2-64) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^2sqrt(x^2-64)

\int x^{2} x^{2} - 64 d x

4096 (\frac{1}{12288} x^{3} x^{2} - 64 - \frac{1}{49152} x^{3} x^{2} - 64 - \frac{1}{512} x x^{2} - 64 - \frac{1}{8} ln (\frac{1}{8} x^{2} - 64 + x)) + C

Solution steps

\int x^{2} x^{2} - 64 d x

Multiply by the conjugate of

x^{2} - 64 : \frac{x ^{4} - 4096}{x ^{2} + 64}

= \int x^{2} \frac{x ^{4} - 4096}{x ^{2} + 64} d x

Apply Integral Substitution

= \int 4096 u^{2} (u + 1) (u - 1) d u

Take the constant out:

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

= 4096 \cdot \int u^{2} (u + 1) (u - 1) d u

Multiply by the conjugate of

u + 1 : \frac{u ^{2} - 1}{u - 1}

= 4096 \cdot \int u^{2} \frac{u ^{2} - 1}{u - 1} (u - 1) d u

Multiply

\frac{u ^{2} - 1}{u - 1} (u - 1) : u^{2} - 1

= 4096 \cdot \int u^{2} u^{2} - 1 d u

Apply Integration By Parts

: \frac{1}{3} u^{3} u^{2} - 1 - \int \frac{u ^{4}}{3 u ^{2} - 1} d u

= 4096 (\frac{1}{3} u^{3} u^{2} - 1 - \int \frac{u ^{4}}{3 u ^{2} - 1} d u)

\int \frac{u ^{4}}{3 u ^{2} - 1} d u = \frac{1}{12} u^{3} u^{2} - 1 + \frac{1}{8} (u u^{2} - 1 + ln u^{2} - 1 + u)

= 4096 (\frac{1}{3} u^{3} u^{2} - 1 - (\frac{1}{12} u^{3} u^{2} - 1 + \frac{1}{8} (u u^{2} - 1 + ln u^{2} - 1 + u)))

Substitute back

u = \frac{x}{8}

= 4096 (\frac{1}{3} (\frac{x}{8})^{3} (\frac{x}{8})^{2} - 1 - (\frac{1}{12} (\frac{x}{8})^{3} (\frac{x}{8})^{2} - 1 + \frac{1}{8} (\frac{x}{8} (\frac{x}{8})^{2} - 1 + ln (\frac{x}{8})^{2} - 1 + \frac{x}{8})))

Simplify

4096 (\frac{1}{3} (\frac{x}{8})^{3} (\frac{x}{8})^{2} - 1 - (\frac{1}{12} (\frac{x}{8})^{3} (\frac{x}{8})^{2} - 1 + \frac{1}{8} (\frac{x}{8} (\frac{x}{8})^{2} - 1 + ln (\frac{x}{8})^{2} - 1 + \frac{x}{8}))) : 4096 (\frac{1}{12288} x^{3} x^{2} - 64 - \frac{1}{49152} x^{3} x^{2} - 64 - \frac{1}{512} x x^{2} - 64 - \frac{1}{8} ln (\frac{1}{8} x^{2} - 64 + x))

= 4096 (\frac{1}{12288} x^{3} x^{2} - 64 - \frac{1}{49152} x^{3} x^{2} - 64 - \frac{1}{512} x x^{2} - 64 - \frac{1}{8} ln (\frac{1}{8} x^{2} - 64 + x))

Add a constant to the solution

= 4096 (\frac{1}{12288} x^{3} x^{2} - 64 - \frac{1}{49152} x^{3} x^{2} - 64 - \frac{1}{512} x x^{2} - 64 - \frac{1}{8} ln (\frac{1}{8} x^{2} - 64 + x)) + C

ln (\frac{2}{x})

n = 1 \sum \infty n \frac{1}{2 ^{n}}

\frac{d}{d x} (π^{x t a n (x)})

g (x) = x arcsinh (x) - 1 + x^{2}

\frac{1}{( s + 3 ) ^{4}}