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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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