What is the integral of (1+2x)/(2-3sqrt(x)) ?

The integral of (1+2x)/(2-3sqrt(x)) is 2(-2/9 x^{3/2}-(2x)/9-17/27 sqrt(x)-34/81 ln|-3sqrt(x)+2|)+C

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integral of (1+2x)/(2-3sqrt(x))

\int \frac{1 + 2 x}{2 - 3 x} d x

2 (- \frac{2}{9} x^{\frac{3}{2}} - \frac{2 x}{9} - \frac{17}{27} x - \frac{34}{81} ln - 3 x + 2) + C

Solution steps

\int \frac{1 + 2 x}{2 - 3 x} d x

Apply u-substitution

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

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

Long division

\frac{u ( 1 + 2 u ^{2} )}{2 - 3 u} : - \frac{2 u ^{2}}{3} - \frac{4 u}{9} - \frac{17}{27} + \frac{34}{27 ( - 3 u + 2 )}

= 2 \cdot \int - \frac{2 u ^{2}}{3} - \frac{4 u}{9} - \frac{17}{27} + \frac{34}{27 ( - 3 u + 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

= 2 (- \int \frac{2 u ^{2}}{3} d u - \int \frac{4 u}{9} d u - \int \frac{17}{27} d u + \int \frac{34}{27 ( - 3 u + 2 )} d u)

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

\int \frac{4 u}{9} d u = \frac{2 u ^{2}}{9}

\int \frac{17}{27} d u = \frac{17}{27} u

\int \frac{34}{27 ( - 3 u + 2 )} d u = - \frac{34}{81} ln ∣ - 3 u + 2 ∣

= 2 (- \frac{2 u ^{3}}{9} - \frac{2 u ^{2}}{9} - \frac{17}{27} u - \frac{34}{81} ln ∣ - 3 u + 2 ∣)

Substitute back

u = x

= 2 (- \frac{2 ( x ) ^{3}}{9} - \frac{2 ( x ) ^{2}}{9} - \frac{17}{27} x - \frac{34}{81} ln - 3 x + 2)

Simplify

2 (- \frac{2 ( x ) ^{3}}{9} - \frac{2 ( x ) ^{2}}{9} - \frac{17}{27} x - \frac{34}{81} ln - 3 x + 2) : 2 (- \frac{2}{9} x^{\frac{3}{2}} - \frac{2 x}{9} - \frac{17}{27} x - \frac{34}{81} ln - 3 x + 2)

= 2 (- \frac{2}{9} x^{\frac{3}{2}} - \frac{2 x}{9} - \frac{17}{27} x - \frac{34}{81} ln - 3 x + 2)

Add a constant to the solution

= 2 (- \frac{2}{9} x^{\frac{3}{2}} - \frac{2 x}{9} - \frac{17}{27} x - \frac{34}{81} ln - 3 x + 2) + C

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What is the integral of (1+2x)/(2-3sqrt(x)) ?
The integral of (1+2x)/(2-3sqrt(x)) is 2(-2/9 x^{3/2}-(2x)/9-17/27 sqrt(x)-34/81 ln|-3sqrt(x)+2|)+C