en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^2(x+3)^{11}

Solution

\int x^{2} (x + 3)^{11} d x

Solution

\frac{x ^{14}}{14} + \frac{33 x ^{13}}{13} + \frac{165 x ^{12}}{4} + 405 x^{11} + 2673 x^{10} + 12474 x^{9} + \frac{168399 x ^{8}}{4} + \frac{721710 x ^{7}}{7} + \frac{360855 x ^{6}}{2} + 216513 x^{5} + \frac{649539 x ^{4}}{4} + 59049 x^{3} + C

Solution steps

\int x^{2} (x + 3)^{11} d x

Expand

x^{2} (x + 3)^{11} : x^{13} + 33 x^{12} + 495 x^{11} + 4455 x^{10} + 26730 x^{9} + 112266 x^{8} + 336798 x^{7} + 721710 x^{6} + 1082565 x^{5} + 1082565 x^{4} + 649539 x^{3} + 177147 x^{2}

= \int x^{13} + 33 x^{12} + 495 x^{11} + 4455 x^{10} + 26730 x^{9} + 112266 x^{8} + 336798 x^{7} + 721710 x^{6} + 1082565 x^{5} + 1082565 x^{4} + 649539 x^{3} + 177147 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^{13} d x + \int 33 x^{12} d x + \int 495 x^{11} d x + \int 4455 x^{10} d x + \int 26730 x^{9} d x + \int 112266 x^{8} d x + \int 336798 x^{7} d x + \int 721710 x^{6} d x + \int 1082565 x^{5} d x + \int 1082565 x^{4} d x + \int 649539 x^{3} d x + \int 177147 x^{2} d x

\int x^{13} d x = \frac{x ^{14}}{14}

\int 33 x^{12} d x = \frac{33 x ^{13}}{13}

\int 495 x^{11} d x = \frac{165 x ^{12}}{4}

\int 4455 x^{10} d x = 405 x^{11}

\int 26730 x^{9} d x = 2673 x^{10}

\int 112266 x^{8} d x = 12474 x^{9}

\int 336798 x^{7} d x = \frac{168399 x ^{8}}{4}

\int 721710 x^{6} d x = \frac{721710 x ^{7}}{7}

\int 1082565 x^{5} d x = \frac{360855 x ^{6}}{2}

\int 1082565 x^{4} d x = 216513 x^{5}

\int 649539 x^{3} d x = \frac{649539 x ^{4}}{4}

\int 177147 x^{2} d x = 59049 x^{3}

= \frac{x ^{14}}{14} + \frac{33 x ^{13}}{13} + \frac{165 x ^{12}}{4} + 405 x^{11} + 2673 x^{10} + 12474 x^{9} + \frac{168399 x ^{8}}{4} + \frac{721710 x ^{7}}{7} + \frac{360855 x ^{6}}{2} + 216513 x^{5} + \frac{649539 x ^{4}}{4} + 59049 x^{3}

Add a constant to the solution

= \frac{x ^{14}}{14} + \frac{33 x ^{13}}{13} + \frac{165 x ^{12}}{4} + 405 x^{11} + 2673 x^{10} + 12474 x^{9} + \frac{168399 x ^{8}}{4} + \frac{721710 x ^{7}}{7} + \frac{360855 x ^{6}}{2} + 216513 x^{5} + \frac{649539 x ^{4}}{4} + 59049 x^{3} + C

Popular Examples

integral of xarctanh(x)

\int x arctanh (x) d x

integral of (3x-7)^3

\int (3 x - 7)^{3} d x

integral of tan^6(5x)

\int tan^{6} (5 x) d x

integral of y^4e^{-y}

\int y^{4} e^{- y} d y

integral of 5/(3a

\int \frac{5}{3 a d} d y

Frequently Asked Questions (FAQ)

What is the integral of x^2(x+3)^{11} ?
The integral of x^2(x+3)^{11} is