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

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

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integral of (2x)/(sqrt(1+2x))

\int \frac{2 x}{1 + 2 x} d x

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

Solution steps

\int \frac{2 x}{1 + 2 x} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{1 + 2 x} d x

Apply u-substitution

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 \cdot \frac{1}{2} (\int u^{2} d u - \int 1 d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

= 2 \cdot \frac{1}{2} (\frac{u ^{3}}{3} - u)

Substitute back

u = 1 + 2 x

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

Simplify

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

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

Add a constant to the solution

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

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