What is the integral of (9x)/(sqrt(2+15x)) ?

The integral of (9x)/(sqrt(2+15x)) is 2/25 (1/3 (2+15x)^{3/2}-2sqrt(2+15x))+C

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integral of (9x)/(sqrt(2+15x))

\int \frac{9 x}{2 + 15 x} d x

\frac{2}{25} (\frac{1}{3} (2 + 15 x)^{\frac{3}{2}} - 2 2 + 15 x) + C

Solution steps

\int \frac{9 x}{2 + 15 x} d x

Take the constant out:

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

= 9 \cdot \int \frac{x}{2 + 15 x} d x

Apply u-substitution

: \int \frac{2 ( u ^{2} - 2 )}{225} d u

= 9 \cdot \int \frac{2 ( u ^{2} - 2 )}{225} d u

Take the constant out:

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

= 9 \cdot \frac{2}{225} \cdot \int u^{2} - 2 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

= 9 \cdot \frac{2}{225} (\int u^{2} d u - \int 2 d u)

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

\int 2 d u = 2 u

= 9 \cdot \frac{2}{225} (\frac{u ^{3}}{3} - 2 u)

Substitute back

u = 2 + 15 x

= 9 \cdot \frac{2}{225} (\frac{( 2 + 15 x ) ^{3}}{3} - 2 2 + 15 x)

Simplify

9 \cdot \frac{2}{225} (\frac{( 2 + 15 x ) ^{3}}{3} - 2 2 + 15 x) : \frac{2}{25} (\frac{1}{3} (2 + 15 x)^{\frac{3}{2}} - 2 2 + 15 x)

= \frac{2}{25} (\frac{1}{3} (2 + 15 x)^{\frac{3}{2}} - 2 2 + 15 x)

Add a constant to the solution

= \frac{2}{25} (\frac{1}{3} (2 + 15 x)^{\frac{3}{2}} - 2 2 + 15 x) + C

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What is the integral of (9x)/(sqrt(2+15x)) ?
The integral of (9x)/(sqrt(2+15x)) is 2/25 (1/3 (2+15x)^{3/2}-2sqrt(2+15x))+C