What is the integral of (2x+1)^5 ?

The integral of (2x+1)^5 is 1/12 (2x+1)^6+C

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integral of (2x+1)^5

\int (2 x + 1)^{5} d x

\frac{1}{12} (2 x + 1)^{6} + C

Solution steps

\int (2 x + 1)^{5} d x

Apply u-substitution

: \int \frac{u ^{5}}{2} d u

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

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{1}{2} \cdot \frac{u ^{6}}{6}

Substitute back

u = 2 x + 1

= \frac{1}{2} \cdot \frac{( 2 x + 1 ) ^{6}}{6}

Simplify

\frac{1}{2} \cdot \frac{( 2 x + 1 ) ^{6}}{6} : \frac{1}{12} (2 x + 1)^{6}

= \frac{1}{12} (2 x + 1)^{6}

Add a constant to the solution

= \frac{1}{12} (2 x + 1)^{6} + C

y^{^{''}} - 2 y^{^{'}} - 3 y = 3 x^{2} - 5

\frac{d y}{d x} = \frac{y ^{2}}{e ^{x}}

y = 1 4^{- \frac{8}{x}}

\frac{d}{d x} (20 + x)

\frac{\partial}{\partial x} ((x y - 3) e^{- x y})