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

The integral of (x+sqrt(2)x)^2 is (-1+sqrt(2))/3 (x+sqrt(2)x)^3+C

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integral of (x+sqrt(2)x)^2

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

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

Solution steps

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

Apply u-substitution

: \int - (1 - 2) u^{2} d u

= \int - (1 - 2) u^{2} d u

Take the constant out:

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

= - (1 - 2) \cdot \int u^{2} d u

Simplify

= (- 1 + 2) \cdot \int u^{2} d u

\int u^{2} d u = \frac{u ^{3}}{3}

= (- 1 + 2) \frac{u ^{3}}{3}

Substitute back

u = x + 2 x

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (x+sqrt(2)x)^2 ?
The integral of (x+sqrt(2)x)^2 is (-1+sqrt(2))/3 (x+sqrt(2)x)^3+C