What is the integral of sqrt(1+(9x^{1/2))^2} ?

The integral of sqrt(1+(9x^{1/2))^2} is 2/243 (1+81x)^{3/2}+C

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integral of sqrt(1+(9x^{1/2))^2}

\int 1 + (9 x^{\frac{1}{2}})^{2} d x

\frac{2}{243} (1 + 81 x)^{\frac{3}{2}} + C

Solution steps

\int 1 + (9 x^{\frac{1}{2}})^{2} d x

Simplify

1 + (9 x^{\frac{1}{2}})^{2} : 1 + 81 x

= \int 1 + 81 x d x

Apply u-substitution

= \int \frac{u}{81} d u

Take the constant out:

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

= \frac{1}{81} \cdot \int u d u

Apply the Power Rule

= \frac{1}{81} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 + 81 x

= \frac{1}{81} \cdot \frac{2}{3} (1 + 81 x)^{\frac{3}{2}}

Simplify

\frac{1}{81} \cdot \frac{2}{3} (1 + 81 x)^{\frac{3}{2}} : \frac{2}{243} (1 + 81 x)^{\frac{3}{2}}

= \frac{2}{243} (1 + 81 x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{243} (1 + 81 x)^{\frac{3}{2}} + C

What is the integral of sqrt(1+(9x^{1/2))^2} ?
The integral of sqrt(1+(9x^{1/2))^2} is 2/243 (1+81x)^{3/2}+C