What is the integral of (50x+10x^3)/(10x^2+x^4) ?

The integral of (50x+10x^3)/(10x^2+x^4) is 5ln(x)+5/2 ln|x^2+10|+C

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integral of (50x+10x^3)/(10x^2+x^4)

\int \frac{50 x + 10 x ^{3}}{10 x ^{2} + x ^{4}} d x

5 ln (x) + \frac{5}{2} ln x^{2} + 10 + C

Solution steps

\int \frac{50 x + 10 x ^{3}}{10 x ^{2} + x ^{4}} d x

Expand

\frac{50 x + 10 x ^{3}}{10 x ^{2} + x ^{4}} : \frac{50 x}{10 x ^{2} + x ^{4}} + \frac{10 x ^{3}}{10 x ^{2} + x ^{4}}

Apply the Sum Rule:

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

= \int \frac{50 x}{10 x ^{2} + x ^{4}} d x + \int \frac{10 x ^{3}}{10 x ^{2} + x ^{4}} d x

\int \frac{50 x}{10 x ^{2} + x ^{4}} d x = 50 (\frac{1}{10} ln (x) - \frac{1}{20} ln x^{2} + 10)

\int \frac{10 x ^{3}}{10 x ^{2} + x ^{4}} d x = 5 ln x^{2} + 10

= 50 (\frac{1}{10} ln (x) - \frac{1}{20} ln x^{2} + 10) + 5 ln x^{2} + 10

Simplify

50 (\frac{1}{10} ln (x) - \frac{1}{20} ln x^{2} + 10) + 5 ln x^{2} + 10 : 5 ln (x) + \frac{5}{2} ln x^{2} + 10

= 5 ln (x) + \frac{5}{2} ln x^{2} + 10

Add a constant to the solution

= 5 ln (x) + \frac{5}{2} ln x^{2} + 10 + C

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What is the integral of (50x+10x^3)/(10x^2+x^4) ?
The integral of (50x+10x^3)/(10x^2+x^4) is 5ln(x)+5/2 ln|x^2+10|+C