What is the integral of sqrt(1+(9x)/4) ?

The integral of sqrt(1+(9x)/4) is 1/27 (4+9x)^{3/2}+C

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integral of sqrt(1+(9x)/4)

\int 1 + \frac{9 x}{4} d x

\frac{1}{27} (4 + 9 x)^{\frac{3}{2}} + C

Solution steps

\int 1 + \frac{9 x}{4} d x

Apply u-substitution

= \int \frac{4 u}{9} d u

Take the constant out:

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

= \frac{4}{9} \cdot \int u d u

Apply the Power Rule

= \frac{4}{9} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + \frac{9 x}{4}

= \frac{4}{9} \cdot \frac{2}{3} (1 + \frac{9 x}{4})^{\frac{3}{2}}

Simplify

\frac{4}{9} \cdot \frac{2}{3} (1 + \frac{9 x}{4})^{\frac{3}{2}} : \frac{1}{27} (4 + 9 x)^{\frac{3}{2}}

= \frac{1}{27} (4 + 9 x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{27} (4 + 9 x)^{\frac{3}{2}} + C

What is the integral of sqrt(1+(9x)/4) ?
The integral of sqrt(1+(9x)/4) is 1/27 (4+9x)^{3/2}+C