What is the integral of 2(x+2)^3 ?

The integral of 2(x+2)^3 is 1/2 (x+2)^4+C

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integral of 2(x+2)^3

\int 2 (x + 2)^{3} d x

\frac{1}{2} (x + 2)^{4} + C

Solution steps

\int 2 (x + 2)^{3} d x

Take the constant out:

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

= 2 \cdot \int (x + 2)^{3} d x

Apply u-substitution

: \int u^{3} d u

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

Apply the Power Rule

: \frac{u ^{4}}{4}

= 2 \cdot \frac{u ^{4}}{4}

Substitute back

u = x + 2

= 2 \cdot \frac{( x + 2 ) ^{4}}{4}

Simplify

2 \cdot \frac{( x + 2 ) ^{4}}{4} : \frac{1}{2} (x + 2)^{4}

= \frac{1}{2} (x + 2)^{4}

Add a constant to the solution

= \frac{1}{2} (x + 2)^{4} + C

\frac{\partial}{\partial x} (x^{3} + 3 x y^{2} - 15 x - 12 y)

\frac{d ^{2} y}{d t ^{2}} - 5 \frac{d y}{d t} - 24 y = 0

x \to 0 - lim (x - 1)

\int_{0}^{1} 3 (x^{2} + 1) e^{- x} d x

x \to 3 + lim (7 x^{2} + 5)