What is the integral of (sqrt(2x+1)) ?

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

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integral of (sqrt(2x+1))

\int (2 x + 1) d x

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

Solution steps

\int 2 x + 1 d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = 2 x + 1

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

Simplify

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

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

Add a constant to the solution

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

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\frac{d}{d x} (x^{- 2 c o s (x)})

x \to 0 lim (\frac{( e ^{x} - e ^{- x} - 2 x )}{1 - sin ( x )})

x \to \infty lim (x \cdot π)

x \to \infty lim (\frac{1}{6})

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