What is the integral of (2x+y)^8 ?

The integral of (2x+y)^8 is 1/18 (y+2x)^9+C

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integral of (2x+y)^8

\int (2 x + y)^{8} d x

\frac{1}{18} (y + 2 x)^{9} + C

Solution steps

\int (2 x + y)^{8} d x

Apply u-substitution

: \int \frac{u ^{8}}{2} d u

= \int \frac{u ^{8}}{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^{8} d u

Apply the Power Rule

: \frac{u ^{9}}{9}

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

Substitute back

u = y + 2 x

= \frac{1}{2} \cdot \frac{( y + 2 x ) ^{9}}{9}

Simplify

\frac{1}{2} \cdot \frac{( y + 2 x ) ^{9}}{9} : \frac{1}{18} (y + 2 x)^{9}

= \frac{1}{18} (y + 2 x)^{9}

Add a constant to the solution

= \frac{1}{18} (y + 2 x)^{9} + C

x \to 0 + lim ((x)^{- x^{2}})

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