What is the integral of (x+2)sqrt(4x+x^2) ?

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

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integral of (x+2)sqrt(4x+x^2)

\int (x + 2) 4 x + x^{2} d x

\frac{1}{3} (4 x + x^{2})^{\frac{3}{2}} + C

Solution steps

\int (x + 2) 4 x + x^{2} d x

Apply u-substitution

= \int u^{2} d u

Apply the Power Rule

= \frac{u ^{3}}{3}

Substitute back

u = 4 x + x^{2}

= \frac{( 4 x + x ^{2} ) ^{3}}{3}

Simplify

\frac{( 4 x + x ^{2} ) ^{3}}{3} : \frac{1}{3} (4 x + x^{2})^{\frac{3}{2}}

= \frac{1}{3} (4 x + x^{2})^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3} (4 x + x^{2})^{\frac{3}{2}} + C

\int \frac{1}{n ^{3}}

x \to \infty lim (3 + \frac{4}{x})

d er i v a t i v e f (x) = (\frac{3}{x})

9 (t + 1) \frac{d y}{d t} - 5 y = 20 t

\int \frac{u}{1 - u} d u

What is the integral of (x+2)sqrt(4x+x^2) ?
The integral of (x+2)sqrt(4x+x^2) is 1/3 (4x+x^2)^{3/2}+C