What is the integral from 0 to 4 of (2x+1)^{1/2} ?

The integral from 0 to 4 of (2x+1)^{1/2} is 26/3

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integral from 0 to 4 of (2x+1)^{1/2}

\int_{0}^{4} (2 x + 1)^{\frac{1}{2}} d x

\frac{26}{3}

Decimal

8.66666 \dots

Solution steps

\int_{0}^{4} (2 x + 1)^{\frac{1}{2}} d x

Apply Integration By Parts

= [x (2 x + 1)^{\frac{1}{2}} - \int \frac{x}{( 2 x + 1 ) ^{\frac{1}{2}}} d x]_{0}^{4}

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

= [x (2 x + 1)^{\frac{1}{2}} - \frac{1}{3} (x - 1) (2 x + 1)^{\frac{1}{2}}]_{0}^{4}

Simplify

[x (2 x + 1)^{\frac{1}{2}} - \frac{1}{3} (x - 1) (2 x + 1)^{\frac{1}{2}}]_{0}^{4} : [\frac{1}{3} (2 x (2 x + 1)^{\frac{1}{2}} + (2 x + 1)^{\frac{1}{2}})]_{0}^{4}

= [\frac{1}{3} (2 x (2 x + 1)^{\frac{1}{2}} + (2 x + 1)^{\frac{1}{2}})]_{0}^{4}

Compute the boundaries:

\frac{26}{3}

= \frac{26}{3}

What is the integral from 0 to 4 of (2x+1)^{1/2} ?
The integral from 0 to 4 of (2x+1)^{1/2} is 26/3