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

The integral of x(x^2+1)^{10} is 1/22 (x^2+1)^{11}+C

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

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

\frac{1}{22} (x^{2} + 1)^{11} + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{10}}{2} d u

= \int \frac{u ^{10}}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int u^{10} d u

Apply the Power Rule

: \frac{u ^{11}}{11}

= \frac{1}{2} \cdot \frac{u ^{11}}{11}

Substitute back

u = x^{2} + 1

= \frac{1}{2} \cdot \frac{( x ^{2} + 1 ) ^{11}}{11}

Simplify

\frac{1}{2} \cdot \frac{( x ^{2} + 1 ) ^{11}}{11} : \frac{1}{22} (x^{2} + 1)^{11}

= \frac{1}{22} (x^{2} + 1)^{11}

Add a constant to the solution

= \frac{1}{22} (x^{2} + 1)^{11} + C

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\frac{d}{d x} (3 x^{2} + \frac{16 x}{3} - 2)

\frac{\partial}{\partial y} (x cos (x y))

What is the integral of x(x^2+1)^{10} ?
The integral of x(x^2+1)^{10} is 1/22 (x^2+1)^{11}+C