What is the integral of (x+3)/(x^2sqrt(2x+3)) ?

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

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integral of (x+3)/(x^2sqrt(2x+3))

\int \frac{x + 3}{x ^{2} 2 x + 3} d x

- \frac{2 x + 3}{x} + C

Solution steps

\int \frac{x + 3}{x ^{2} 2 x + 3} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply Integral Substitution

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

Take the constant out:

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

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

Factor

\frac{v ^{2} + 1}{( v ^{2} - 1 ) ^{2}}

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

Apply u-substitution

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

Take the partial fraction of

\frac{w ^{2} - 2 w + 2}{w ^{2} ( w - 2 ) ^{2}} : \frac{1}{2 w ^{2}} + \frac{1}{2 ( w - 2 ) ^{2}}

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

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}{3} (\int \frac{1}{2 w ^{2}} d w + \int \frac{1}{2 ( w - 2 ) ^{2}} d w)

\int \frac{1}{2 w ^{2}} d w = - \frac{1}{2 w}

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

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

Substitute back

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

Simplify

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

= - \frac{2 x + 3}{x}

Add a constant to the solution

= - \frac{2 x + 3}{x} + C

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