What is the integral of 2sqrt(2x+3) ?

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

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integral of 2sqrt(2x+3)

\int 2 2 x + 3 d x

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

Solution steps

\int 2 2 x + 3 d x

Take the constant out:

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

= 2 \cdot \int 2 x + 3 d x

Apply u-substitution

: \int \frac{u}{2} d u

= 2 \cdot \int \frac{u}{2} d u

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = 2 x + 3

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

Simplify

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

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

Add a constant to the solution

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

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