What is the integral of (10)/(5x^2-2x^3) ?

The integral of (10)/(5x^2-2x^3) is 10(-1/25 (-2ln|x|+5/x)-2/25 ln|2x-5|)+C

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integral of (10)/(5x^2-2x^3)

\int \frac{10}{5 x ^{2} - 2 x ^{3}} d x

10 (- \frac{1}{25} (- 2 ln ∣ x ∣ + \frac{5}{x}) - \frac{2}{25} ln ∣ 2 x - 5 ∣) + C

Solution steps

\int \frac{10}{5 x ^{2} - 2 x ^{3}} d x

Take the constant out:

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

= 10 \cdot \int \frac{1}{5 x ^{2} - 2 x ^{3}} d x

Take the partial fraction of

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

= 10 \cdot \int - \frac{- 2 x - 5}{25 x ^{2}} - \frac{4}{25 ( 2 x - 5 )} d x

Apply the Sum Rule:

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

= 10 (- \int \frac{- 2 x - 5}{25 x ^{2}} d x - \int \frac{4}{25 ( 2 x - 5 )} d x)

\int \frac{- 2 x - 5}{25 x ^{2}} d x = \frac{1}{25} (- 2 ln ∣ x ∣ + \frac{5}{x})

\int \frac{4}{25 ( 2 x - 5 )} d x = \frac{2}{25} ln ∣ 2 x - 5 ∣

= 10 (- \frac{1}{25} (- 2 ln ∣ x ∣ + \frac{5}{x}) - \frac{2}{25} ln ∣ 2 x - 5 ∣)

Add a constant to the solution

= 10 (- \frac{1}{25} (- 2 ln ∣ x ∣ + \frac{5}{x}) - \frac{2}{25} ln ∣ 2 x - 5 ∣) + C

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What is the integral of (10)/(5x^2-2x^3) ?
The integral of (10)/(5x^2-2x^3) is 10(-1/25 (-2ln|x|+5/x)-2/25 ln|2x-5|)+C