en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (1+sqrt(x))^8

Solution

\int (1 + x)^{8} d x

Solution

x + \frac{16}{3} x^{\frac{3}{2}} + 14 x^{2} + \frac{112}{5} x^{\frac{5}{2}} + \frac{70 x ^{3}}{3} + 16 x^{\frac{7}{2}} + 7 x^{4} + \frac{16}{9} x^{\frac{9}{2}} + \frac{x ^{5}}{5} + C

Solution steps

\int (1 + x)^{8} d x

Expand

(1 + x)^{8} : 1 + 8 x + 28 x + 56 x^{\frac{3}{2}} + 70 x^{2} + 56 x^{\frac{5}{2}} + 28 x^{3} + 8 x^{\frac{7}{2}} + x^{4}

= \int 1 + 8 x + 28 x + 56 x^{\frac{3}{2}} + 70 x^{2} + 56 x^{\frac{5}{2}} + 28 x^{3} + 8 x^{\frac{7}{2}} + x^{4} 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 1 d x + \int 8 x d x + \int 28 x d x + \int 56 x^{\frac{3}{2}} d x + \int 70 x^{2} d x + \int 56 x^{\frac{5}{2}} d x + \int 28 x^{3} d x + \int 8 x^{\frac{7}{2}} d x + \int x^{4} d x

\int 1 d x = x

\int 8 x d x = \frac{16}{3} x^{\frac{3}{2}}

\int 28 x d x = 14 x^{2}

\int 56 x^{\frac{3}{2}} d x = \frac{112}{5} x^{\frac{5}{2}}

\int 70 x^{2} d x = \frac{70 x ^{3}}{3}

\int 56 x^{\frac{5}{2}} d x = 16 x^{\frac{7}{2}}

\int 28 x^{3} d x = 7 x^{4}

\int 8 x^{\frac{7}{2}} d x = \frac{16}{9} x^{\frac{9}{2}}

\int x^{4} d x = \frac{x ^{5}}{5}

= x + \frac{16}{3} x^{\frac{3}{2}} + 14 x^{2} + \frac{112}{5} x^{\frac{5}{2}} + \frac{70 x ^{3}}{3} + 16 x^{\frac{7}{2}} + 7 x^{4} + \frac{16}{9} x^{\frac{9}{2}} + \frac{x ^{5}}{5}

Add a constant to the solution

= x + \frac{16}{3} x^{\frac{3}{2}} + 14 x^{2} + \frac{112}{5} x^{\frac{5}{2}} + \frac{70 x ^{3}}{3} + 16 x^{\frac{7}{2}} + 7 x^{4} + \frac{16}{9} x^{\frac{9}{2}} + \frac{x ^{5}}{5} + C

Graph

Popular Examples

tangent of (7x)/(x-1)tangent of

\frac{7 x}{x - 1}

integral from 0 to infinity of xe^{-xy}

\int_{0}^{\infty} x e^{- x y} d x

derivative of (sqrt(x))(3-x)^2 derivative of

(x) (3 - x)^{2}

derivative of ycos(xln(y))

\frac{d}{d x} (y cos (x ln (y)))

simplify x/(2-sqrt(x))simplify

\frac{x}{2 - x}

Frequently Asked Questions (FAQ)

What is the integral of (1+sqrt(x))^8 ?
The integral of (1+sqrt(x))^8 is