What is the integral of (1+sqrt(x))^{14} ?

The integral of (1+sqrt(x))^{14} is

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integral of (1+sqrt(x))^{14}

\int (1 + x)^{14} d x

x + \frac{28}{3} x^{\frac{3}{2}} + \frac{91 x ^{2}}{2} + \frac{728}{5} x^{\frac{5}{2}} + \frac{1001 x ^{3}}{3} + 572 x^{\frac{7}{2}} + \frac{3003 x ^{4}}{4} + \frac{2288}{3} x^{\frac{9}{2}} + \frac{3003 x ^{5}}{5} + 364 x^{\frac{11}{2}} + \frac{1001 x ^{6}}{6} + 56 x^{\frac{13}{2}} + 13 x^{7} + \frac{28}{15} x^{\frac{15}{2}} + \frac{x ^{8}}{8} + C

Solution steps

\int (1 + x)^{14} d x

Expand

(1 + x)^{14} : 1 + 14 x + 91 x + 364 x^{\frac{3}{2}} + 1001 x^{2} + 2002 x^{\frac{5}{2}} + 3003 x^{3} + 3432 x^{\frac{7}{2}} + 3003 x^{4} + 2002 x^{\frac{9}{2}} + 1001 x^{5} + 364 x^{\frac{11}{2}} + 91 x^{6} + 14 x^{\frac{13}{2}} + x^{7}

= \int 1 + 14 x + 91 x + 364 x^{\frac{3}{2}} + 1001 x^{2} + 2002 x^{\frac{5}{2}} + 3003 x^{3} + 3432 x^{\frac{7}{2}} + 3003 x^{4} + 2002 x^{\frac{9}{2}} + 1001 x^{5} + 364 x^{\frac{11}{2}} + 91 x^{6} + 14 x^{\frac{13}{2}} + x^{7} 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 14 x d x + \int 91 x d x + \int 364 x^{\frac{3}{2}} d x + \int 1001 x^{2} d x + \int 2002 x^{\frac{5}{2}} d x + \int 3003 x^{3} d x + \int 3432 x^{\frac{7}{2}} d x + \int 3003 x^{4} d x + \int 2002 x^{\frac{9}{2}} d x + \int 1001 x^{5} d x + \int 364 x^{\frac{11}{2}} d x + \int 91 x^{6} d x + \int 14 x^{\frac{13}{2}} d x + \int x^{7} d x

\int 1 d x = x

\int 14 x d x = \frac{28}{3} x^{\frac{3}{2}}

\int 91 x d x = \frac{91 x ^{2}}{2}

\int 364 x^{\frac{3}{2}} d x = \frac{728}{5} x^{\frac{5}{2}}

\int 1001 x^{2} d x = \frac{1001 x ^{3}}{3}

\int 2002 x^{\frac{5}{2}} d x = 572 x^{\frac{7}{2}}

\int 3003 x^{3} d x = \frac{3003 x ^{4}}{4}

\int 3432 x^{\frac{7}{2}} d x = \frac{2288}{3} x^{\frac{9}{2}}

\int 3003 x^{4} d x = \frac{3003 x ^{5}}{5}

\int 2002 x^{\frac{9}{2}} d x = 364 x^{\frac{11}{2}}

\int 1001 x^{5} d x = \frac{1001 x ^{6}}{6}

\int 364 x^{\frac{11}{2}} d x = 56 x^{\frac{13}{2}}

\int 91 x^{6} d x = 13 x^{7}

\int 14 x^{\frac{13}{2}} d x = \frac{28}{15} x^{\frac{15}{2}}

\int x^{7} d x = \frac{x ^{8}}{8}

= x + \frac{28}{3} x^{\frac{3}{2}} + \frac{91 x ^{2}}{2} + \frac{728}{5} x^{\frac{5}{2}} + \frac{1001 x ^{3}}{3} + 572 x^{\frac{7}{2}} + \frac{3003 x ^{4}}{4} + \frac{2288}{3} x^{\frac{9}{2}} + \frac{3003 x ^{5}}{5} + 364 x^{\frac{11}{2}} + \frac{1001 x ^{6}}{6} + 56 x^{\frac{13}{2}} + 13 x^{7} + \frac{28}{15} x^{\frac{15}{2}} + \frac{x ^{8}}{8}

Add a constant to the solution

= x + \frac{28}{3} x^{\frac{3}{2}} + \frac{91 x ^{2}}{2} + \frac{728}{5} x^{\frac{5}{2}} + \frac{1001 x ^{3}}{3} + 572 x^{\frac{7}{2}} + \frac{3003 x ^{4}}{4} + \frac{2288}{3} x^{\frac{9}{2}} + \frac{3003 x ^{5}}{5} + 364 x^{\frac{11}{2}} + \frac{1001 x ^{6}}{6} + 56 x^{\frac{13}{2}} + 13 x^{7} + \frac{28}{15} x^{\frac{15}{2}} + \frac{x ^{8}}{8} + C

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