en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x(x^2-4)^8

Solution

\int x (x^{2} - 4)^{8} d x

Solution

\frac{1}{2} (x^{2} (x^{2} - 4)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 4)^{7} - \frac{1}{18} (7 x^{18} - 189 x^{16} + 2160 x^{14} - 13440 x^{12} + 48384 x^{10} - 96768 x^{8} + 86016 x^{6}))) + C

Solution steps

\int x (x^{2} - 4)^{8} d x

Apply u-substitution

: \int \frac{( u - 4 ) ^{8}}{2} d u

= \int \frac{( u - 4 ) ^{8}}{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 (u - 4)^{8} d u

Apply Integration By Parts

: (u - 4)^{8} u - \int 8 (u - 4)^{7} u d u

= \frac{1}{2} ((u - 4)^{8} u - \int 8 (u - 4)^{7} u d u)

\int 8 (u - 4)^{7} u d u = 8 (\frac{1}{2} u^{2} (u - 4)^{7} - \frac{1}{18} (7 u^{9} - 189 u^{8} + 2160 u^{7} - 13440 u^{6} + 48384 u^{5} - 96768 u^{4} + 86016 u^{3}))

= \frac{1}{2} ((u - 4)^{8} u - 8 (\frac{1}{2} u^{2} (u - 4)^{7} - \frac{1}{18} (7 u^{9} - 189 u^{8} + 2160 u^{7} - 13440 u^{6} + 48384 u^{5} - 96768 u^{4} + 86016 u^{3})))

Substitute back

u = x^{2}

= \frac{1}{2} ((x^{2} - 4)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 4)^{7} - \frac{1}{18} (7 (x^{2})^{9} - 189 (x^{2})^{8} + 2160 (x^{2})^{7} - 13440 (x^{2})^{6} + 48384 (x^{2})^{5} - 96768 (x^{2})^{4} + 86016 (x^{2})^{3})))

Simplify

\frac{1}{2} ((x^{2} - 4)^{8} x^{2} - 8 (\frac{1}{2} (x^{2})^{2} (x^{2} - 4)^{7} - \frac{1}{18} (7 (x^{2})^{9} - 189 (x^{2})^{8} + 2160 (x^{2})^{7} - 13440 (x^{2})^{6} + 48384 (x^{2})^{5} - 96768 (x^{2})^{4} + 86016 (x^{2})^{3}))) : \frac{1}{2} (x^{2} (x^{2} - 4)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 4)^{7} - \frac{1}{18} (7 x^{18} - 189 x^{16} + 2160 x^{14} - 13440 x^{12} + 48384 x^{10} - 96768 x^{8} + 86016 x^{6})))

= \frac{1}{2} (x^{2} (x^{2} - 4)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 4)^{7} - \frac{1}{18} (7 x^{18} - 189 x^{16} + 2160 x^{14} - 13440 x^{12} + 48384 x^{10} - 96768 x^{8} + 86016 x^{6})))

Add a constant to the solution

= \frac{1}{2} (x^{2} (x^{2} - 4)^{8} - 8 (\frac{1}{2} x^{4} (x^{2} - 4)^{7} - \frac{1}{18} (7 x^{18} - 189 x^{16} + 2160 x^{14} - 13440 x^{12} + 48384 x^{10} - 96768 x^{8} + 86016 x^{6}))) + C

Popular Examples

integral of sin^2(x)sin^2(2x)

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

integral of ((e^x))/(e^x+4)

\int \frac{( e ^{x} )}{e ^{x} + 4} d x

\int 2 e d x

integral of ((3x^4-x^3+6x^2))/(x^4)

\int \frac{( 3 x ^{4} - x ^{3} + 6 x ^{2} )}{x ^{4}} d x

integral of e^{(t^2)/4}

\int e^{\frac{t ^{2}}{4}} d t

Frequently Asked Questions (FAQ)

What is the integral of x(x^2-4)^8 ?
The integral of x(x^2-4)^8 is