What is the integral of (x+1)(x-2)^9 ?

The integral of (x+1)(x-2)^9 is 1/11 (x-2)^{11}+3/10 (x-2)^{10}+C

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integral of (x+1)(x-2)^9

\int (x + 1) (x - 2)^{9} d x

\frac{1}{11} (x - 2)^{11} + \frac{3}{10} (x - 2)^{10} + C

Solution steps

\int (x + 1) (x - 2)^{9} d x

Apply u-substitution

: \int u (u - 3)^{9} d u

= \int u (u - 3)^{9} d u

Apply u-substitution

: \int (v + 3) v^{9} d v

= \int (v + 3) v^{9} d v

Expand

(v + 3) v^{9} : v^{10} + 3 v^{9}

= \int v^{10} + 3 v^{9} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int v^{10} d v + \int 3 v^{9} d v

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

\int 3 v^{9} d v = \frac{3 v ^{10}}{10}

= \frac{v ^{11}}{11} + \frac{3 v ^{10}}{10}

Substitute back

= \frac{( x + 1 - 3 ) ^{11}}{11} + \frac{3 ( x + 1 - 3 ) ^{10}}{10}

Simplify

\frac{( x + 1 - 3 ) ^{11}}{11} + \frac{3 ( x + 1 - 3 ) ^{10}}{10} : \frac{1}{11} (x - 2)^{11} + \frac{3}{10} (x - 2)^{10}

= \frac{1}{11} (x - 2)^{11} + \frac{3}{10} (x - 2)^{10}

Add a constant to the solution

= \frac{1}{11} (x - 2)^{11} + \frac{3}{10} (x - 2)^{10} + C

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