What is the integral of (2x)/(sqrt(3-5x)) ?

The integral of (2x)/(sqrt(3-5x)) is 4/25 (1/3 (3-5x)^{3/2}-3sqrt(3-5x))+C

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integral of (2x)/(sqrt(3-5x))

\int \frac{2 x}{3 - 5 x} d x

\frac{4}{25} (\frac{1}{3} (3 - 5 x)^{\frac{3}{2}} - 3 3 - 5 x) + C

Solution steps

\int \frac{2 x}{3 - 5 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}{3 - 5 x} d x

Apply u-substitution

: \int \frac{2 ( u ^{2} - 3 )}{25} d u

= 2 \cdot \int \frac{2 ( u ^{2} - 3 )}{25} d u

Take the constant out:

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

= 2 \cdot \frac{2}{25} \cdot \int u^{2} - 3 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{2}{25} (\int u^{2} d u - \int 3 d u)

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

\int 3 d u = 3 u

= 2 \cdot \frac{2}{25} (\frac{u ^{3}}{3} - 3 u)

Substitute back

u = 3 - 5 x

= 2 \cdot \frac{2}{25} (\frac{( 3 - 5 x ) ^{3}}{3} - 3 3 - 5 x)

Simplify

2 \cdot \frac{2}{25} (\frac{( 3 - 5 x ) ^{3}}{3} - 3 3 - 5 x) : \frac{4}{25} (\frac{1}{3} (3 - 5 x)^{\frac{3}{2}} - 3 3 - 5 x)

= \frac{4}{25} (\frac{1}{3} (3 - 5 x)^{\frac{3}{2}} - 3 3 - 5 x)

Add a constant to the solution

= \frac{4}{25} (\frac{1}{3} (3 - 5 x)^{\frac{3}{2}} - 3 3 - 5 x) + C

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What is the integral of (2x)/(sqrt(3-5x)) ?
The integral of (2x)/(sqrt(3-5x)) is 4/25 (1/3 (3-5x)^{3/2}-3sqrt(3-5x))+C