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

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

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integral of sqrt(1+(3/2 x^{1/2))^2}

\int 1 + (\frac{3}{2} x^{\frac{1}{2}})^{2} d x

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

Solution steps

\int 1 + (\frac{3}{2} x^{\frac{1}{2}})^{2} d x

Simplify

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

= \int \frac{1}{2} (4 + 9 x)^{\frac{1}{2}} d x

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u ^{\frac{1}{2}}}{9} d u

= \frac{1}{2} \cdot \int \frac{u ^{\frac{1}{2}}}{9} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{9} \cdot \int u^{\frac{1}{2}} d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{2} \cdot \frac{1}{9} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 4 + 9 x

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

Simplify

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

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