What is the integral of (5x+2)^3 ?

The integral of (5x+2)^3 is 1/20 (5x+2)^4+C

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integral of (5x+2)^3

\int (5 x + 2)^{3} d x

\frac{1}{20} (5 x + 2)^{4} + C

Solution steps

\int (5 x + 2)^{3} d x

Apply u-substitution

: \int \frac{u ^{3}}{5} d u

= \int \frac{u ^{3}}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int u^{3} d u

Apply the Power Rule

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

= \frac{1}{5} \cdot \frac{u ^{4}}{4}

Substitute back

u = 5 x + 2

= \frac{1}{5} \cdot \frac{( 5 x + 2 ) ^{4}}{4}

Simplify

\frac{1}{5} \cdot \frac{( 5 x + 2 ) ^{4}}{4} : \frac{1}{20} (5 x + 2)^{4}

= \frac{1}{20} (5 x + 2)^{4}

Add a constant to the solution

= \frac{1}{20} (5 x + 2)^{4} + C

y^{^{''}} + y = e^{x}

\frac{d y}{d x} = \frac{( x ^{2} - 1 )}{2 y}, y (3) = 1

\frac{\partial}{\partial y} (y^{2} e^{x yz})

\frac{\partial}{\partial x} (y^{5} - 4 x y)

y^{^{'}} = \frac{6 y ^{3}}{( 3 - 2 x ) ^{2}}, y (0) = - 1