What is the integral of (3sqrt(x))/(x+9) ?

The integral of (3sqrt(x))/(x+9) is 18(-arctan(1/3 sqrt(x))+1/3 sqrt(x))+C

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integral of (3sqrt(x))/(x+9)

\int \frac{3 x}{x + 9} d x

18 (- arctan (\frac{1}{3} x) + \frac{1}{3} x) + C

Solution steps

\int \frac{3 x}{x + 9} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{x + 9} d x

Apply u-substitution

: \int \frac{2 u ^{2}}{u ^{2} + 9} d u

= 3 \cdot \int \frac{2 u ^{2}}{u ^{2} + 9} d u

Take the constant out:

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

= 3 \cdot 2 \cdot \int \frac{u ^{2}}{u ^{2} + 9} d u

\frac{u ^{2}}{u ^{2} + 9} = - \frac{9}{u ^{2} + 9} + 1

= 3 \cdot 2 \cdot \int - \frac{9}{u ^{2} + 9} + 1 d u

Apply Integral Substitution

= 3 \cdot 2 \cdot \int 3 (- \frac{1}{v ^{2} + 1} + 1) d v

Take the constant out:

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

= 3 \cdot 2 \cdot 3 \cdot \int - \frac{1}{v ^{2} + 1} + 1 d v

Apply the Sum Rule:

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

= 3 \cdot 2 \cdot 3 (- \int \frac{1}{v ^{2} + 1} d v + \int 1 d v)

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

\int 1 d v = v

= 3 \cdot 2 \cdot 3 (- arctan (v) + v)

Substitute back

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

Simplify

3 \cdot 2 \cdot 3 (- arctan (\frac{x}{3}) + \frac{x}{3}) : 18 (- arctan (\frac{1}{3} x) + \frac{1}{3} x)

= 18 (- arctan (\frac{1}{3} x) + \frac{1}{3} x)

Add a constant to the solution

= 18 (- arctan (\frac{1}{3} x) + \frac{1}{3} x) + C

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What is the integral of (3sqrt(x))/(x+9) ?
The integral of (3sqrt(x))/(x+9) is 18(-arctan(1/3 sqrt(x))+1/3 sqrt(x))+C