What is the integral of x(x+2)^7 ?

The integral of x(x+2)^7 is (x^9)/9+(7x^8)/4+12x^7+(140x^6)/3+112x^5+168x^4+(448x^3)/3+64x^2+C

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)^7

\int x (x + 2)^{7} d x

\frac{x ^{9}}{9} + \frac{7 x ^{8}}{4} + 12 x^{7} + \frac{140 x ^{6}}{3} + 112 x^{5} + 168 x^{4} + \frac{448 x ^{3}}{3} + 64 x^{2} + C

Solution steps

\int x (x + 2)^{7} d x

Expand

x (x + 2)^{7} : x^{8} + 14 x^{7} + 84 x^{6} + 280 x^{5} + 560 x^{4} + 672 x^{3} + 448 x^{2} + 128 x

= \int x^{8} + 14 x^{7} + 84 x^{6} + 280 x^{5} + 560 x^{4} + 672 x^{3} + 448 x^{2} + 128 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{8} d x + \int 14 x^{7} d x + \int 84 x^{6} d x + \int 280 x^{5} d x + \int 560 x^{4} d x + \int 672 x^{3} d x + \int 448 x^{2} d x + \int 128 x d x

\int x^{8} d x = \frac{x ^{9}}{9}

\int 14 x^{7} d x = \frac{7 x ^{8}}{4}

\int 84 x^{6} d x = 12 x^{7}

\int 280 x^{5} d x = \frac{140 x ^{6}}{3}

\int 560 x^{4} d x = 112 x^{5}

\int 672 x^{3} d x = 168 x^{4}

\int 448 x^{2} d x = \frac{448 x ^{3}}{3}

\int 128 x d x = 64 x^{2}

= \frac{x ^{9}}{9} + \frac{7 x ^{8}}{4} + 12 x^{7} + \frac{140 x ^{6}}{3} + 112 x^{5} + 168 x^{4} + \frac{448 x ^{3}}{3} + 64 x^{2}

Add a constant to the solution

= \frac{x ^{9}}{9} + \frac{7 x ^{8}}{4} + 12 x^{7} + \frac{140 x ^{6}}{3} + 112 x^{5} + 168 x^{4} + \frac{448 x ^{3}}{3} + 64 x^{2} + C

\int x^{2} \cdot arcsin (x) d x

\int \frac{1}{t ^{2} t ^{2} - 4} d t

\int sin (6 π) t d t

\int \frac{y}{e ^{2 y^{2}}} d y

\int \frac{3 x}{x ^{2} - 2 x - 8} d x

What is the integral of x(x+2)^7 ?
The integral of x(x+2)^7 is (x^9)/9+(7x^8)/4+12x^7+(140x^6)/3+112x^5+168x^4+(448x^3)/3+64x^2+C