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+1)^{10}

Solution

\int (x^{2} + 1)^{10} d x

Solution

\frac{x ^{21}}{21} + \frac{10 x ^{19}}{19} + \frac{45 x ^{17}}{17} + 8 x^{15} + \frac{210 x ^{13}}{13} + \frac{252 x ^{11}}{11} + \frac{70 x ^{9}}{3} + \frac{120 x ^{7}}{7} + 9 x^{5} + \frac{10 x ^{3}}{3} + x + C

Solution steps

\int (x^{2} + 1)^{10} d x

Expand

(x^{2} + 1)^{10} : x^{20} + 10 x^{18} + 45 x^{16} + 120 x^{14} + 210 x^{12} + 252 x^{10} + 210 x^{8} + 120 x^{6} + 45 x^{4} + 10 x^{2} + 1

= \int x^{20} + 10 x^{18} + 45 x^{16} + 120 x^{14} + 210 x^{12} + 252 x^{10} + 210 x^{8} + 120 x^{6} + 45 x^{4} + 10 x^{2} + 1 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^{20} d x + \int 10 x^{18} d x + \int 45 x^{16} d x + \int 120 x^{14} d x + \int 210 x^{12} d x + \int 252 x^{10} d x + \int 210 x^{8} d x + \int 120 x^{6} d x + \int 45 x^{4} d x + \int 10 x^{2} d x + \int 1 d x

\int x^{20} d x = \frac{x ^{21}}{21}

\int 10 x^{18} d x = \frac{10 x ^{19}}{19}

\int 45 x^{16} d x = \frac{45 x ^{17}}{17}

\int 120 x^{14} d x = 8 x^{15}

\int 210 x^{12} d x = \frac{210 x ^{13}}{13}

\int 252 x^{10} d x = \frac{252 x ^{11}}{11}

\int 210 x^{8} d x = \frac{70 x ^{9}}{3}

\int 120 x^{6} d x = \frac{120 x ^{7}}{7}

\int 45 x^{4} d x = 9 x^{5}

\int 10 x^{2} d x = \frac{10 x ^{3}}{3}

\int 1 d x = x

= \frac{x ^{21}}{21} + \frac{10 x ^{19}}{19} + \frac{45 x ^{17}}{17} + 8 x^{15} + \frac{210 x ^{13}}{13} + \frac{252 x ^{11}}{11} + \frac{70 x ^{9}}{3} + \frac{120 x ^{7}}{7} + 9 x^{5} + \frac{10 x ^{3}}{3} + x

Add a constant to the solution

= \frac{x ^{21}}{21} + \frac{10 x ^{19}}{19} + \frac{45 x ^{17}}{17} + 8 x^{15} + \frac{210 x ^{13}}{13} + \frac{252 x ^{11}}{11} + \frac{70 x ^{9}}{3} + \frac{120 x ^{7}}{7} + 9 x^{5} + \frac{10 x ^{3}}{3} + x + C

Graph

Popular Examples

integral of (-1)/(t^2)

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

integral of (e^{sqrt(ln(x))})/(2xsqrt(ln(x)))

\int \frac{e ^{l n (x)}}{2 x ln ( x )} d x

integral of (-ln(x))^2

\int (- ln (x))^{2} d x

integral of t/(sin^2(t))

\int \frac{t}{sin ^{2} ( t )} d t

integral of (cosh(ln(x)))/x

\int \frac{cosh ( ln ( x ) )}{x} d x

Frequently Asked Questions (FAQ)

What is the integral of (x^2+1)^{10} ?
The integral of (x^2+1)^{10} is