en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^3+1)^8x^2

Solution

\int (x^{3} + 1)^{8} x^{2} d x

Solution

\frac{x ^{27}}{27} + \frac{x ^{24}}{3} + \frac{4 x ^{21}}{3} + \frac{28 x ^{18}}{9} + \frac{14 x ^{15}}{3} + \frac{14 x ^{12}}{3} + \frac{28 x ^{9}}{9} + \frac{4 x ^{6}}{3} + \frac{x ^{3}}{3} + C

Solution steps

\int (x^{3} + 1)^{8} x^{2} d x

Expand

(x^{3} + 1)^{8} x^{2} : x^{26} + 8 x^{23} + 28 x^{20} + 56 x^{17} + 70 x^{14} + 56 x^{11} + 28 x^{8} + 8 x^{5} + x^{2}

= \int x^{26} + 8 x^{23} + 28 x^{20} + 56 x^{17} + 70 x^{14} + 56 x^{11} + 28 x^{8} + 8 x^{5} + x^{2} 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^{26} d x + \int 8 x^{23} d x + \int 28 x^{20} d x + \int 56 x^{17} d x + \int 70 x^{14} d x + \int 56 x^{11} d x + \int 28 x^{8} d x + \int 8 x^{5} d x + \int x^{2} d x

\int x^{26} d x = \frac{x ^{27}}{27}

\int 8 x^{23} d x = \frac{x ^{24}}{3}

\int 28 x^{20} d x = \frac{4 x ^{21}}{3}

\int 56 x^{17} d x = \frac{28 x ^{18}}{9}

\int 70 x^{14} d x = \frac{14 x ^{15}}{3}

\int 56 x^{11} d x = \frac{14 x ^{12}}{3}

\int 28 x^{8} d x = \frac{28 x ^{9}}{9}

\int 8 x^{5} d x = \frac{4 x ^{6}}{3}

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

= \frac{x ^{27}}{27} + \frac{x ^{24}}{3} + \frac{4 x ^{21}}{3} + \frac{28 x ^{18}}{9} + \frac{14 x ^{15}}{3} + \frac{14 x ^{12}}{3} + \frac{28 x ^{9}}{9} + \frac{4 x ^{6}}{3} + \frac{x ^{3}}{3}

Add a constant to the solution

= \frac{x ^{27}}{27} + \frac{x ^{24}}{3} + \frac{4 x ^{21}}{3} + \frac{28 x ^{18}}{9} + \frac{14 x ^{15}}{3} + \frac{14 x ^{12}}{3} + \frac{28 x ^{9}}{9} + \frac{4 x ^{6}}{3} + \frac{x ^{3}}{3} + C

Popular Examples

integral of 1/(sqrt(3+4x-4x^2))

\int \frac{1}{3 + 4 x - 4 x ^{2}} d x

integral of x^nln(ax)

\int x^{n} ln (a x) d x

integral of sqrt(10^{3t)}

\int 1 0^{3 t} d t

integral of 1/n cos(nx)

\int \frac{1}{n} cos (n x) d x

integral of-2x^2e^{-x^2}

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

Frequently Asked Questions (FAQ)

What is the integral of (x^3+1)^8x^2 ?
The integral of (x^3+1)^8x^2 is