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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int \frac{u ^{9}}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int u^{9} d u

Apply the Power Rule

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

Substitute back

u = x^{2} + 1

= 2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 1 ) ^{10}}{10}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 1 ) ^{10}}{10} : \frac{1}{10} (x^{2} + 1)^{10}

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

Add a constant to the solution

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

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