What is the integral of (x-1)^{10}(x+3)^2 ?

The integral of (x-1)^{10}(x+3)^2 is

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integral of (x-1)^{10}(x+3)^2

\int (x - 1)^{10} (x + 3)^{2} d x

\frac{x ^{13}}{13} - \frac{x ^{12}}{3} - \frac{6 x ^{11}}{11} + 6 x^{10} - \frac{35 x ^{9}}{3} - 9 x^{8} + 84 x^{7} - 188 x^{6} + 243 x^{5} - 205 x^{4} + \frac{346 x ^{3}}{3} - 42 x^{2} + 9 x + C

Solution steps

\int (x - 1)^{10} (x + 3)^{2} d x

Expand

(x - 1)^{10} (x + 3)^{2} : x^{12} - 4 x^{11} - 6 x^{10} + 60 x^{9} - 105 x^{8} - 72 x^{7} + 588 x^{6} - 1128 x^{5} + 1215 x^{4} - 820 x^{3} + 346 x^{2} - 84 x + 9

= \int x^{12} - 4 x^{11} - 6 x^{10} + 60 x^{9} - 105 x^{8} - 72 x^{7} + 588 x^{6} - 1128 x^{5} + 1215 x^{4} - 820 x^{3} + 346 x^{2} - 84 x + 9 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^{12} d x - \int 4 x^{11} d x - \int 6 x^{10} d x + \int 60 x^{9} d x - \int 105 x^{8} d x - \int 72 x^{7} d x + \int 588 x^{6} d x - \int 1128 x^{5} d x + \int 1215 x^{4} d x - \int 820 x^{3} d x + \int 346 x^{2} d x - \int 84 x d x + \int 9 d x

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

\int 4 x^{11} d x = \frac{x ^{12}}{3}

\int 6 x^{10} d x = \frac{6 x ^{11}}{11}

\int 60 x^{9} d x = 6 x^{10}

\int 105 x^{8} d x = \frac{35 x ^{9}}{3}

\int 72 x^{7} d x = 9 x^{8}

\int 588 x^{6} d x = 84 x^{7}

\int 1128 x^{5} d x = 188 x^{6}

\int 1215 x^{4} d x = 243 x^{5}

\int 820 x^{3} d x = 205 x^{4}

\int 346 x^{2} d x = \frac{346 x ^{3}}{3}

\int 84 x d x = 42 x^{2}

\int 9 d x = 9 x

= \frac{x ^{13}}{13} - \frac{x ^{12}}{3} - \frac{6 x ^{11}}{11} + 6 x^{10} - \frac{35 x ^{9}}{3} - 9 x^{8} + 84 x^{7} - 188 x^{6} + 243 x^{5} - 205 x^{4} + \frac{346 x ^{3}}{3} - 42 x^{2} + 9 x

Add a constant to the solution

= \frac{x ^{13}}{13} - \frac{x ^{12}}{3} - \frac{6 x ^{11}}{11} + 6 x^{10} - \frac{35 x ^{9}}{3} - 9 x^{8} + 84 x^{7} - 188 x^{6} + 243 x^{5} - 205 x^{4} + \frac{346 x ^{3}}{3} - 42 x^{2} + 9 x + C

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

\frac{d}{d x} (\frac{3}{sin ( x ) + cos ( x )})

x \to 0 lim (cot (2 x))

\int (x - 5) sin (π x) d x

What is the integral of (x-1)^{10}(x+3)^2 ?
The integral of (x-1)^{10}(x+3)^2 is