What is the integral of x^2*sqrt(x^2-9) ?

The integral of x^2*sqrt(x^2-9) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^2*sqrt(x^2-9)

\int x^{2} \cdot x^{2} - 9 d x

81 (\frac{1}{243} x^{3} x^{2} - 9 - \frac{1}{972} x^{3} x^{2} - 9 - \frac{1}{72} x x^{2} - 9 - \frac{1}{8} ln (\frac{1}{3} x^{2} - 9 + x)) + C

Solution steps

\int x^{2} x^{2} - 9 d x

Multiply by the conjugate of

x^{2} - 9 : \frac{x ^{4} - 81}{x ^{2} + 9}

= \int x^{2} \frac{x ^{4} - 81}{x ^{2} + 9} d x

Apply Integral Substitution

= \int 81 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

= 81 \cdot \int u^{2} (u + 1) (u - 1) d u

Multiply by the conjugate of

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

= 81 \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

= 81 \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

= 81 (\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)

= 81 (\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}{3}

= 81 (\frac{1}{3} (\frac{x}{3})^{3} (\frac{x}{3})^{2} - 1 - (\frac{1}{12} (\frac{x}{3})^{3} (\frac{x}{3})^{2} - 1 + \frac{1}{8} (\frac{x}{3} (\frac{x}{3})^{2} - 1 + ln (\frac{x}{3})^{2} - 1 + \frac{x}{3})))

Simplify

81 (\frac{1}{3} (\frac{x}{3})^{3} (\frac{x}{3})^{2} - 1 - (\frac{1}{12} (\frac{x}{3})^{3} (\frac{x}{3})^{2} - 1 + \frac{1}{8} (\frac{x}{3} (\frac{x}{3})^{2} - 1 + ln (\frac{x}{3})^{2} - 1 + \frac{x}{3}))) : 81 (\frac{1}{243} x^{3} x^{2} - 9 - \frac{1}{972} x^{3} x^{2} - 9 - \frac{1}{72} x x^{2} - 9 - \frac{1}{8} ln (\frac{1}{3} x^{2} - 9 + x))

= 81 (\frac{1}{243} x^{3} x^{2} - 9 - \frac{1}{972} x^{3} x^{2} - 9 - \frac{1}{72} x x^{2} - 9 - \frac{1}{8} ln (\frac{1}{3} x^{2} - 9 + x))

Add a constant to the solution

= 81 (\frac{1}{243} x^{3} x^{2} - 9 - \frac{1}{972} x^{3} x^{2} - 9 - \frac{1}{72} x x^{2} - 9 - \frac{1}{8} ln (\frac{1}{3} x^{2} - 9 + x)) + C

\frac{d P}{d t} = 0.5 (P) (1 - \frac{P}{1000})

\frac{d y}{d x} = \frac{3 sec ( y )}{( x + 1 ) ^{2}}

f (x) = \frac{4 x}{( x ^{2} + 1 ) ^{2}}

n = 1 \sum \infty n (sin (nπ))

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