What is the integral of (x^2+1)^5 ?

The integral of (x^2+1)^5 is (x^{11}}{11}+\frac{5x^9)/9+(10x^7)/7+2x^5+(5x^3)/3+x+C

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integral of (x^2+1)^5

\int (x^{2} + 1)^{5} d x

\frac{x ^{11}}{11} + \frac{5 x ^{9}}{9} + \frac{10 x ^{7}}{7} + 2 x^{5} + \frac{5 x ^{3}}{3} + x + C

Solution steps

\int (x^{2} + 1)^{5} d x

Expand

(x^{2} + 1)^{5} : x^{10} + 5 x^{8} + 10 x^{6} + 10 x^{4} + 5 x^{2} + 1

= \int x^{10} + 5 x^{8} + 10 x^{6} + 10 x^{4} + 5 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^{10} d x + \int 5 x^{8} d x + \int 10 x^{6} d x + \int 10 x^{4} d x + \int 5 x^{2} d x + \int 1 d x

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

\int 5 x^{8} d x = \frac{5 x ^{9}}{9}

\int 10 x^{6} d x = \frac{10 x ^{7}}{7}

\int 10 x^{4} d x = 2 x^{5}

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

\int 1 d x = x

= \frac{x ^{11}}{11} + \frac{5 x ^{9}}{9} + \frac{10 x ^{7}}{7} + 2 x^{5} + \frac{5 x ^{3}}{3} + x

Add a constant to the solution

= \frac{x ^{11}}{11} + \frac{5 x ^{9}}{9} + \frac{10 x ^{7}}{7} + 2 x^{5} + \frac{5 x ^{3}}{3} + x + C

\int + \frac{cos ( 3 x )}{1 + sin ^{2} ( 3 x )} d x

\frac{d}{d x} (sin (3 x - 2))

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

\int \frac{62 x ^{9}}{2835} d x

x \to 1 + lim (\frac{1}{x ^{2} - 4 x + 3})

What is the integral of (x^2+1)^5 ?
The integral of (x^2+1)^5 is (x^{11}}{11}+\frac{5x^9)/9+(10x^7)/7+2x^5+(5x^3)/3+x+C