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

The integral of x(x+1)^{10} is ((x+1)^{12})/(12)-((x+1)^{11})/(11)+C

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integral of x(x+1)^{10}

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

\frac{( x + 1 ) ^{12}}{12} - \frac{( x + 1 ) ^{11}}{11} + C

Solution steps

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

Apply u-substitution

: \int (u - 1) u^{10} d u

= \int (u - 1) u^{10} d u

Expand

(u - 1) u^{10} : u^{11} - u^{10}

= \int u^{11} - u^{10} d u

Apply the Sum Rule:

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

= \int u^{11} d u - \int u^{10} d u

\int u^{11} d u = \frac{u ^{12}}{12}

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

= \frac{u ^{12}}{12} - \frac{u ^{11}}{11}

Substitute back

u = x + 1

= \frac{( x + 1 ) ^{12}}{12} - \frac{( x + 1 ) ^{11}}{11}

Add a constant to the solution

= \frac{( x + 1 ) ^{12}}{12} - \frac{( x + 1 ) ^{11}}{11} + C

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What is the integral of x(x+1)^{10} ?
The integral of x(x+1)^{10} is ((x+1)^{12})/(12)-((x+1)^{11})/(11)+C